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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Xie%2C+W&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Xie%2C+W&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Xie%2C+W&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.09428">arXiv:2502.09428</a> <span> [<a href="https://arxiv.org/pdf/2502.09428">pdf</a>, <a href="https://arxiv.org/format/2502.09428">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Multicontinuum Modeling of Time-Fractional Diffusion-Wave Equation in Heterogeneous Media </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bai%2C+H">Huiran Bai</a>, <a href="/search/math?searchtype=author&query=Ammosov%2C+D">Dmitry Ammosov</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yin Yang</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wei Xie</a>, <a href="/search/math?searchtype=author&query=Kobaisi%2C+M+A">Mohammed Al Kobaisi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.09428v1-abstract-short" style="display: inline;"> This paper considers a time-fractional diffusion-wave equation with a high-contrast heterogeneous diffusion coefficient. A numerical solution to this problem can present great computational challenges due to its multiscale nature. Therefore, in this paper, we derive a multicontinuum time-fractional diffusion-wave model using the multicontinuum homogenization method. For this purpose, we formulate… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.09428v1-abstract-full').style.display = 'inline'; document.getElementById('2502.09428v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.09428v1-abstract-full" style="display: none;"> This paper considers a time-fractional diffusion-wave equation with a high-contrast heterogeneous diffusion coefficient. A numerical solution to this problem can present great computational challenges due to its multiscale nature. Therefore, in this paper, we derive a multicontinuum time-fractional diffusion-wave model using the multicontinuum homogenization method. For this purpose, we formulate constraint cell problems considering various homogenized effects. These cell problems are implemented in oversampled regions to avoid boundary effects. By solving the cell problems, we obtain multicontinuum expansions of fine-scale solutions. Then, using these multicontinuum expansions and supposing the smoothness of the macroscopic variables, we rigorously derive the corresponding multicontinuum model. Finally, we present numerical results for two-dimensional model problems with different time-fractional derivatives to verify the accuracy of our proposed approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.09428v1-abstract-full').style.display = 'none'; document.getElementById('2502.09428v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35B27 (Primary) 26A33; 65M60 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.07529">arXiv:2502.07529</a> <span> [<a href="https://arxiv.org/pdf/2502.07529">pdf</a>, <a href="https://arxiv.org/format/2502.07529">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Training Deep Learning Models with Norm-Constrained LMOs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Pethick%2C+T">Thomas Pethick</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wanyun Xie</a>, <a href="/search/math?searchtype=author&query=Antonakopoulos%2C+K">Kimon Antonakopoulos</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+Z">Zhenyu Zhu</a>, <a href="/search/math?searchtype=author&query=Silveti-Falls%2C+A">Antonio Silveti-Falls</a>, <a href="/search/math?searchtype=author&query=Cevher%2C+V">Volkan Cevher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.07529v1-abstract-short" style="display: inline;"> In this work, we study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new stochastic family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems. The resulting update rule unifies several existing optimization methods under a single framework… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07529v1-abstract-full').style.display = 'inline'; document.getElementById('2502.07529v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.07529v1-abstract-full" style="display: none;"> In this work, we study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new stochastic family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems. The resulting update rule unifies several existing optimization methods under a single framework. Furthermore, we propose an explicit choice of norm for deep architectures, which, as a side benefit, leads to the transferability of hyperparameters across model sizes. Experimentally, we demonstrate significant speedups on nanoGPT training without any reliance on Adam. The proposed method is memory-efficient, requiring only one set of model weights and one set of gradients, which can be stored in half-precision. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07529v1-abstract-full').style.display = 'none'; document.getElementById('2502.07529v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.08743">arXiv:2501.08743</a> <span> [<a href="https://arxiv.org/pdf/2501.08743">pdf</a>, <a href="https://arxiv.org/ps/2501.08743">ps</a>, <a href="https://arxiv.org/format/2501.08743">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The Global Sections of Chiral de Rham Complexes on Closed Complex Curves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Song%2C+B">Bailin Song</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wujie Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.08743v2-abstract-short" style="display: inline;"> The space of global sections of the chiral de Rham complex on any closed complex curve with genus $g \ge2$ is calculated. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.08743v2-abstract-full" style="display: none;"> The space of global sections of the chiral de Rham complex on any closed complex curve with genus $g \ge2$ is calculated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.08743v2-abstract-full').style.display = 'none'; document.getElementById('2501.08743v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.01229">arXiv:2411.01229</a> <span> [<a href="https://arxiv.org/pdf/2411.01229">pdf</a>, <a href="https://arxiv.org/ps/2411.01229">ps</a>, <a href="https://arxiv.org/format/2411.01229">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> On the ReLU Lagrangian Cuts for Stochastic Mixed Integer Programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Deng%2C+H">Haoyun Deng</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.01229v2-abstract-short" style="display: inline;"> We study stochastic mixed integer programs with both first-stage and recourse decisions involving mixed integer variables. A new family of Lagrangian cuts, termed ``ReLU Lagrangian cuts," is introduced by reformulating the nonanticipativity constraints using ReLU functions. These cuts can be integrated into scenario decomposition methods. We show that including ReLU Lagrangian cuts is sufficient t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01229v2-abstract-full').style.display = 'inline'; document.getElementById('2411.01229v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.01229v2-abstract-full" style="display: none;"> We study stochastic mixed integer programs with both first-stage and recourse decisions involving mixed integer variables. A new family of Lagrangian cuts, termed ``ReLU Lagrangian cuts," is introduced by reformulating the nonanticipativity constraints using ReLU functions. These cuts can be integrated into scenario decomposition methods. We show that including ReLU Lagrangian cuts is sufficient to achieve optimality in the original stochastic mixed integer programs. Without solving the Lagrangian dual problems, we derive closed-form expressions for these cuts. Furthermore, to speed up the cut-generating procedures, we introduce linear programming-based methods to enhance the cut coefficients. Numerical studies demonstrate the effectiveness of the proposed cuts compared to existing cut families. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01229v2-abstract-full').style.display = 'none'; document.getElementById('2411.01229v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.04032">arXiv:2409.04032</a> <span> [<a href="https://arxiv.org/pdf/2409.04032">pdf</a>, <a href="https://arxiv.org/ps/2409.04032">ps</a>, <a href="https://arxiv.org/format/2409.04032">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Double star arrangement and the pointed multinet </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yongqiang Liu</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wentao Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.04032v1-abstract-short" style="display: inline;"> Let $\mathcal{A}$ be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of $\mathcal{A}$. By the work of Falk and Yuzvinsky \cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04032v1-abstract-full').style.display = 'inline'; document.getElementById('2409.04032v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.04032v1-abstract-full" style="display: none;"> Let $\mathcal{A}$ be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of $\mathcal{A}$. By the work of Falk and Yuzvinsky \cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determined. Denham and Suciu introduced the pointed multinet structure to obtain examples of arrangements with translated positive-dimensional components in the degree one cohomology jump loci \cite{DS}. Suciu asked the question if all translated positive-dimensional components appear in this manner \cite{Suc14}. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara and Yoshinaga \cite[Example 3.2]{ISY22} gives a negative answer to this question. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04032v1-abstract-full').style.display = 'none'; document.getElementById('2409.04032v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages, 2 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.08738">arXiv:2408.08738</a> <span> [<a href="https://arxiv.org/pdf/2408.08738">pdf</a>, <a href="https://arxiv.org/format/2408.08738">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> The Blessing of Strategic Customers in Personalized Pricing </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Z">Zhi Chen</a>, <a href="/search/math?searchtype=author&query=Sturt%2C+B">Bradley Sturt</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.08738v1-abstract-short" style="display: inline;"> We consider a feature-based personalized pricing problem in which the buyer is strategic: given the seller's pricing policy, the buyer can augment the features that they reveal to the seller to obtain a low price for the product. We model the seller's pricing problem as a stochastic program over an infinite-dimensional space of pricing policies where the radii by which the buyer can perturb the fe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.08738v1-abstract-full').style.display = 'inline'; document.getElementById('2408.08738v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.08738v1-abstract-full" style="display: none;"> We consider a feature-based personalized pricing problem in which the buyer is strategic: given the seller's pricing policy, the buyer can augment the features that they reveal to the seller to obtain a low price for the product. We model the seller's pricing problem as a stochastic program over an infinite-dimensional space of pricing policies where the radii by which the buyer can perturb the features are strictly positive. We establish that the sample average approximation of this problem is asymptotically consistent; that is, we prove that the objective value of the sample average approximation converges almost surely to the objective value of the stochastic problem as the number of samples tends to infinity under mild technical assumptions. This consistency guarantee thus shows that incorporating strategic consumer behavior into a data-driven pricing problem can, in addition to making the pricing problem more realistic, also help prevent overfitting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.08738v1-abstract-full').style.display = 'none'; document.getElementById('2408.08738v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.17471">arXiv:2404.17471</a> <span> [<a href="https://arxiv.org/pdf/2404.17471">pdf</a>, <a href="https://arxiv.org/format/2404.17471">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Multicontinuum homogenization in perforated domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wei Xie</a>, <a href="/search/math?searchtype=author&query=Efendiev%2C+Y">Yalchin Efendiev</a>, <a href="/search/math?searchtype=author&query=Huang%2C+Y">Yunqing Huang</a>, <a href="/search/math?searchtype=author&query=Leung%2C+W+T">Wing Tat Leung</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yin Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.17471v1-abstract-short" style="display: inline;"> In this paper, we develop a general framework for multicontinuum homogenization in perforated domains. The simulations of problems in perforated domains are expensive and, in many applications, coarse-grid macroscopic models are developed. Many previous approaches include homogenization, multiscale finite element methods, and so on. In our paper, we design multicontinuum homogenization based on ou… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.17471v1-abstract-full').style.display = 'inline'; document.getElementById('2404.17471v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.17471v1-abstract-full" style="display: none;"> In this paper, we develop a general framework for multicontinuum homogenization in perforated domains. The simulations of problems in perforated domains are expensive and, in many applications, coarse-grid macroscopic models are developed. Many previous approaches include homogenization, multiscale finite element methods, and so on. In our paper, we design multicontinuum homogenization based on our recently proposed framework. In this setting, we distinguish different spatial regions in perforations based on their sizes. For example, very thin perforations are considered as one continua, while larger perforations are considered as another continua. By differentiating perforations in this way, we are able to predict flows in each of them more accurately. We present a framework by formulating cell problems for each continuum using appropriate constraints for the solution averages and their gradients. These cell problem solutions are used in a multiscale expansion and in deriving novel macroscopic systems for multicontinuum homogenization. Our proposed approaches are designed for problems without scale separation. We present numerical results for two continuum problems and demonstrate the accuracy of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.17471v1-abstract-full').style.display = 'none'; document.getElementById('2404.17471v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.17372">arXiv:2404.17372</a> <span> [<a href="https://arxiv.org/pdf/2404.17372">pdf</a>, <a href="https://arxiv.org/format/2404.17372">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> CEM-GMsFEM for Poisson equations in heterogeneous perforated domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wei Xie</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yin Yang</a>, <a href="/search/math?searchtype=author&query=Chung%2C+E">Eric Chung</a>, <a href="/search/math?searchtype=author&query=Huang%2C+Y">Yunqing Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.17372v1-abstract-short" style="display: inline;"> In this paper, we propose a novel multiscale model reduction strategy tailored to address the Poisson equation within heterogeneous perforated domains. The numerical simulation of this intricate problem is impeded by its multiscale characteristics, necessitating an exceptionally fine mesh to adequately capture all relevant details. To overcome the challenges inherent in the multiscale nature of th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.17372v1-abstract-full').style.display = 'inline'; document.getElementById('2404.17372v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.17372v1-abstract-full" style="display: none;"> In this paper, we propose a novel multiscale model reduction strategy tailored to address the Poisson equation within heterogeneous perforated domains. The numerical simulation of this intricate problem is impeded by its multiscale characteristics, necessitating an exceptionally fine mesh to adequately capture all relevant details. To overcome the challenges inherent in the multiscale nature of the perforations, we introduce a coarse space constructed using the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). This involves constructing basis functions through a sequence of local energy minimization problems over eigenspaces containing localized information pertaining to the heterogeneities. Through our analysis, we demonstrate that the oversampling layers depend on the local eigenvalues, thereby implicating the local geometry as well. Additionally, we provide numerical examples to illustrate the efficacy of the proposed scheme. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.17372v1-abstract-full').style.display = 'none'; document.getElementById('2404.17372v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.04406">arXiv:2402.04406</a> <span> [<a href="https://arxiv.org/pdf/2402.04406">pdf</a>, <a href="https://arxiv.org/format/2402.04406">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Regularized MIP Model for Integrating Energy Storage Systems and its Application for Solving a Trilevel Interdiction Problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+D">Dahye Han</a>, <a href="/search/math?searchtype=author&query=Jiang%2C+N">Nan Jiang</a>, <a href="/search/math?searchtype=author&query=Dey%2C+S+S">Santanu S. Dey</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.04406v2-abstract-short" style="display: inline;"> Incorporating energy storage systems (ESS) into power systems has been studied in many recent works, where binary variables are often introduced to model the complementary nature of battery charging and discharging. A conventional approach for these ESS optimization problems is to relax binary variables and convert the problem into a linear program. However, such linear programming relaxation mode… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.04406v2-abstract-full').style.display = 'inline'; document.getElementById('2402.04406v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.04406v2-abstract-full" style="display: none;"> Incorporating energy storage systems (ESS) into power systems has been studied in many recent works, where binary variables are often introduced to model the complementary nature of battery charging and discharging. A conventional approach for these ESS optimization problems is to relax binary variables and convert the problem into a linear program. However, such linear programming relaxation models can yield unrealistic fractional solutions, such as simultaneous charging and discharging. In this paper, we develop a regularized Mixed-Integer Programming (MIP) model for the ESS optimal power flow (OPF) problem. We prove that under mild conditions, the proposed regularized model admits a zero integrality gap with its linear programming relaxation; hence, it can be solved efficiently. By studying the properties of the regularized MIP model, we show that its optimal solution is also near-optimal to the original ESS OPF problem, thereby providing a valid and tight upper bound for the ESS OPF problem. The use of the regularized MIP model allows us to solve a trilevel min-max-min network contingency problem which is otherwise intractable to solve. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.04406v2-abstract-full').style.display = 'none'; document.getElementById('2402.04406v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.01872">arXiv:2402.01872</a> <span> [<a href="https://arxiv.org/pdf/2402.01872">pdf</a>, <a href="https://arxiv.org/format/2402.01872">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Distributionally Fair Stochastic Optimization using Wasserstein Distance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+Q">Qing Ye</a>, <a href="/search/math?searchtype=author&query=Hanasusanto%2C+G+A">Grani A. Hanasusanto</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.01872v3-abstract-short" style="display: inline;"> A traditional stochastic program under a finite population typically seeks to optimize efficiency by maximizing the expected profits or minimizing the expected costs, subject to a set of constraints. However, implementing such optimization-based decisions can have varying impacts on individuals, and when assessed using the individuals' utility functions, these impacts may differ substantially acro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.01872v3-abstract-full').style.display = 'inline'; document.getElementById('2402.01872v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.01872v3-abstract-full" style="display: none;"> A traditional stochastic program under a finite population typically seeks to optimize efficiency by maximizing the expected profits or minimizing the expected costs, subject to a set of constraints. However, implementing such optimization-based decisions can have varying impacts on individuals, and when assessed using the individuals' utility functions, these impacts may differ substantially across demographic groups delineated by sensitive attributes, such as gender, race, age, and socioeconomic status. As each group comprises multiple individuals, a common remedy is to enforce group fairness, which necessitates the measurement of disparities in the distributions of utilities across different groups. This paper introduces the concept of Distributionally Fair Stochastic Optimization (DFSO) based on the Wasserstein fairness measure. The DFSO aims to minimize distributional disparities among groups, quantified by the Wasserstein distance, while adhering to an acceptable level of inefficiency. Our analysis reveals that: (i) the Wasserstein fairness measure recovers the demographic parity fairness prevalent in binary classification literature; (ii) this measure can approximate the well-known Kolmogorov-Smirnov fairness measure with considerable accuracy; and (iii) despite DFSO's biconvex nature, the epigraph of the Wasserstein fairness measure is generally Mixed-Integer Convex Programming Representable (MICP-R). Additionally, we introduce two distinct lower bounds for the Wasserstein fairness measure: the Jensen bound, applicable to the general Wasserstein fairness measure, and the Gelbrich bound, specific to the type-2 Wasserstein fairness measure. We establish the exactness of the Gelbrich bound and quantify the theoretical difference between the Wasserstein fairness measure and the Gelbrich bound. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.01872v3-abstract-full').style.display = 'none'; document.getElementById('2402.01872v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">No more revision</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.17899">arXiv:2401.17899</a> <span> [<a href="https://arxiv.org/pdf/2401.17899">pdf</a>, <a href="https://arxiv.org/format/2401.17899">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> On Tractability, Complexity, and Mixed-Integer Convex Programming Representability of Distributionally Favorable Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+N">Nan Jiang</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.17899v1-abstract-short" style="display: inline;"> Distributionally Favorable Optimization (DFO) is an important framework for decision-making under uncertainty, with applications across fields such as reinforcement learning, online learning, robust statistics, chance-constrained programming, and two-stage stochastic optimization without relatively complete recourse. In contrast to the traditional Distributionally Robust Optimization (DRO) paradig… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.17899v1-abstract-full').style.display = 'inline'; document.getElementById('2401.17899v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.17899v1-abstract-full" style="display: none;"> Distributionally Favorable Optimization (DFO) is an important framework for decision-making under uncertainty, with applications across fields such as reinforcement learning, online learning, robust statistics, chance-constrained programming, and two-stage stochastic optimization without relatively complete recourse. In contrast to the traditional Distributionally Robust Optimization (DRO) paradigm, DFO presents a unique challenge -- the application of the inner infimum operator often fails to retain the convexity. In light of this challenge, we study the tractability and complexity of DFO. We establish sufficient and necessary conditions for determining when DFO problems are tractable or intractable. Despite the typical nonconvex nature of DFO problems, our findings show that they are mixed-integer convex programming representable (MICP-R), thereby enabling solutions via standard optimization solvers. Finally, we numerically validate the efficacy of our MICP-R formulations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.17899v1-abstract-full').style.display = 'none'; document.getElementById('2401.17899v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.00308">arXiv:2401.00308</a> <span> [<a href="https://arxiv.org/pdf/2401.00308">pdf</a>, <a href="https://arxiv.org/ps/2401.00308">ps</a>, <a href="https://arxiv.org/format/2401.00308">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> On Sparse Canonical Correlation Analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Dey%2C+S+S">Santanu S. Dey</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.00308v1-abstract-short" style="display: inline;"> The classical Canonical Correlation Analysis (CCA) identifies the correlations between two sets of multivariate variables based on their covariance, which has been widely applied in diverse fields such as computer vision, natural language processing, and speech analysis. Despite its popularity, CCA can encounter challenges in explaining correlations between two variable sets within high-dimensiona… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.00308v1-abstract-full').style.display = 'inline'; document.getElementById('2401.00308v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.00308v1-abstract-full" style="display: none;"> The classical Canonical Correlation Analysis (CCA) identifies the correlations between two sets of multivariate variables based on their covariance, which has been widely applied in diverse fields such as computer vision, natural language processing, and speech analysis. Despite its popularity, CCA can encounter challenges in explaining correlations between two variable sets within high-dimensional data contexts. Thus, this paper studies Sparse Canonical Correlation Analysis (SCCA) that enhances the interpretability of CCA. We first show that SCCA generalizes three well-known sparse optimization problems, sparse PCA, sparse SVD, and sparse regression, which are all classified as NP-hard problems. This result motivates us to develop strong formulations and efficient algorithms. Our main contributions include (i) the introduction of a combinatorial formulation that captures the essence of SCCA and allows the development of approximation algorithms; (ii) the derivation of an equivalent mixed-integer semidefinite programming model that facilitates a specialized branch-and-cut algorithm with analytical cuts; and (iii) the establishment of the complexity results for two low-rank special cases of SCCA. The effectiveness of our proposed formulations and algorithms is validated through numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.00308v1-abstract-full').style.display = 'none'; document.getElementById('2401.00308v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.13173">arXiv:2312.13173</a> <span> [<a href="https://arxiv.org/pdf/2312.13173">pdf</a>, <a href="https://arxiv.org/format/2312.13173">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Learning Fair Policies for Multi-stage Selection Problems from Observational Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jia%2C+Z">Zhuangzhuang Jia</a>, <a href="/search/math?searchtype=author&query=Hanasusanto%2C+G+A">Grani A. Hanasusanto</a>, <a href="/search/math?searchtype=author&query=Vayanos%2C+P">Phebe Vayanos</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.13173v1-abstract-short" style="display: inline;"> We consider the problem of learning fair policies for multi-stage selection problems from observational data. This problem arises in several high-stakes domains such as company hiring, loan approval, or bail decisions where outcomes (e.g., career success, loan repayment, recidivism) are only observed for those selected. We propose a multi-stage framework that can be augmented with various fairness… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13173v1-abstract-full').style.display = 'inline'; document.getElementById('2312.13173v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.13173v1-abstract-full" style="display: none;"> We consider the problem of learning fair policies for multi-stage selection problems from observational data. This problem arises in several high-stakes domains such as company hiring, loan approval, or bail decisions where outcomes (e.g., career success, loan repayment, recidivism) are only observed for those selected. We propose a multi-stage framework that can be augmented with various fairness constraints, such as demographic parity or equal opportunity. This problem is a highly intractable infinite chance-constrained program involving the unknown joint distribution of covariates and outcomes. Motivated by the potential impact of selection decisions on people's lives and livelihoods, we propose to focus on interpretable linear selection rules. Leveraging tools from causal inference and sample average approximation, we obtain an asymptotically consistent solution to this selection problem by solving a mixed binary conic optimization problem, which can be solved using standard off-the-shelf solvers. We conduct extensive computational experiments on a variety of datasets adapted from the UCI repository on which we show that our proposed approaches can achieve an 11.6% improvement in precision and a 38% reduction in the measure of unfairness compared to the existing selection policy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13173v1-abstract-full').style.display = 'none'; document.getElementById('2312.13173v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">38th Annual AAAI Conference on Artificial Intelligence, 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.10939">arXiv:2312.10939</a> <span> [<a href="https://arxiv.org/pdf/2312.10939">pdf</a>, <a href="https://arxiv.org/ps/2312.10939">ps</a>, <a href="https://arxiv.org/format/2312.10939">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> The homology groups of finite cyclic covering of line arrangement complement </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yongqiang Liu</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wentao Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.10939v1-abstract-short" style="display: inline;"> In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams for complexified line arrangemen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.10939v1-abstract-full').style.display = 'inline'; document.getElementById('2312.10939v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.10939v1-abstract-full" style="display: none;"> In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams for complexified line arrangement to any complex line arrangement. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.10939v1-abstract-full').style.display = 'none'; document.getElementById('2312.10939v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 52C35; 32S55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.01418">arXiv:2311.01418</a> <span> [<a href="https://arxiv.org/pdf/2311.01418">pdf</a>, <a href="https://arxiv.org/ps/2311.01418">ps</a>, <a href="https://arxiv.org/format/2311.01418">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Torsion energy with boundary mean zero condition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Q">Qinfeng Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weihong Xie</a>, <a href="/search/math?searchtype=author&query=Yang%2C+H">Hang Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.01418v3-abstract-short" style="display: inline;"> Motivated by establishing Neumann Talenti type comparison results, we concern the minimization of the following shape functional under volume constraint: \begin{align*} T(惟):=\inf\left\{\frac12 \int_惟 |\nabla u|^2\,dx -\int_惟u\,dx: u\in H^1(惟),\ \int_{\partial 惟}ud蟽=0 \right\}. \end{align*} We prove that ball is a local minimizer to $T(\cdot)$ under smooth perturbation, but quite surprisingl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.01418v3-abstract-full').style.display = 'inline'; document.getElementById('2311.01418v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.01418v3-abstract-full" style="display: none;"> Motivated by establishing Neumann Talenti type comparison results, we concern the minimization of the following shape functional under volume constraint: \begin{align*} T(惟):=\inf\left\{\frac12 \int_惟 |\nabla u|^2\,dx -\int_惟u\,dx: u\in H^1(惟),\ \int_{\partial 惟}ud蟽=0 \right\}. \end{align*} We prove that ball is a local minimizer to $T(\cdot)$ under smooth perturbation, but quite surprisingly, ball is not locally minimal to $T(\cdot)$ under Lipschitz perturbation. In fact, let $P_N$ be the regular polygon in $\mathbb{R}^2$ with $N$ sides and area $蟺$, then we prove that $T(P_N)$ is a strictly increasing function with respect to $N$ and $\lim_{N\rightarrow \infty}T(P_N)=T(B)$ where $B$ is the unit disk. As another side result, we prove that in dimension bigger than or equal to three, rigidity results of Serrin's seminal overdetermined system is not stable under Dirichlet perturbations, in contrast to the stability of rigidity under Neumann perturbation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.01418v3-abstract-full').style.display = 'none'; document.getElementById('2311.01418v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.13459">arXiv:2310.13459</a> <span> [<a href="https://arxiv.org/pdf/2310.13459">pdf</a>, <a href="https://arxiv.org/format/2310.13459">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Stable Nonconvex-Nonconcave Training via Linear Interpolation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Pethick%2C+T">Thomas Pethick</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wanyun Xie</a>, <a href="/search/math?searchtype=author&query=Cevher%2C+V">Volkan Cevher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.13459v4-abstract-short" style="display: inline;"> This paper presents a theoretical analysis of linear interpolation as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear interpolation can help by leveraging the theory of nonexpansive operators. We construct a new optimization scheme cal… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.13459v4-abstract-full').style.display = 'inline'; document.getElementById('2310.13459v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.13459v4-abstract-full" style="display: none;"> This paper presents a theoretical analysis of linear interpolation as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear interpolation can help by leveraging the theory of nonexpansive operators. We construct a new optimization scheme called relaxed approximate proximal point (RAPP), which is the first explicit method without anchoring to achieve last iterate convergence rates for $蟻$-comonotone problems while only requiring $蟻> -\tfrac{1}{2L}$. The construction extends to constrained and regularized settings. By replacing the inner optimizer in RAPP we rediscover the family of Lookahead algorithms for which we establish convergence in cohypomonotone problems even when the base optimizer is taken to be gradient descent ascent. The range of cohypomonotone problems in which Lookahead converges is further expanded by exploiting that Lookahead inherits the properties of the base optimizer. We corroborate the results with experiments on generative adversarial networks which demonstrates the benefits of the linear interpolation present in both RAPP and Lookahead. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.13459v4-abstract-full').style.display = 'none'; document.getElementById('2310.13459v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.07564">arXiv:2308.07564</a> <span> [<a href="https://arxiv.org/pdf/2308.07564">pdf</a>, <a href="https://arxiv.org/format/2308.07564">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Fluid Dynamics">physics.flu-dyn</span> </div> </div> <p class="title is-5 mathjax"> MSAT: Matrix stability analysis tool for shock-capturing schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ren%2C+W">Weijie Ren</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wenjia Xie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Ye Zhang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+H">Hang Yu</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Z">Zhengyu Tian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.07564v1-abstract-short" style="display: inline;"> The simulation of supersonic or hypersonic flows often suffers from numerical shock instabilities if the flow field contains strong shocks, limiting the further application of shock-capturing schemes. In this paper, we develop the unified matrix stability analysis method for schemes with three-point stencils and present MSAT, an open-source tool to quantitatively analyze the shock instability prob… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.07564v1-abstract-full').style.display = 'inline'; document.getElementById('2308.07564v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.07564v1-abstract-full" style="display: none;"> The simulation of supersonic or hypersonic flows often suffers from numerical shock instabilities if the flow field contains strong shocks, limiting the further application of shock-capturing schemes. In this paper, we develop the unified matrix stability analysis method for schemes with three-point stencils and present MSAT, an open-source tool to quantitatively analyze the shock instability problem. Based on the finite-volume approach on the structured grid, MSAT can be employed to investigate the mechanism of the shock instability problem, evaluate the robustness of numerical schemes, and then help to develop robust schemes. Also, MSAT has the ability to analyze the practical simulation of supersonic or hypersonic flows, evaluate whether it will suffer from shock instabilities, and then assist in selecting appropriate numerical schemes accordingly. As a result, MSAT is a helpful tool that can investigate the shock instability problem and help to cure it. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.07564v1-abstract-full').style.display = 'none'; document.getElementById('2308.07564v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 6 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.03428">arXiv:2308.03428</a> <span> [<a href="https://arxiv.org/pdf/2308.03428">pdf</a>, <a href="https://arxiv.org/format/2308.03428">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Fluid Dynamics">physics.flu-dyn</span> </div> </div> <p class="title is-5 mathjax"> Numerical stability analysis of shock-capturing methods for strong shocks II: high-order finite-volume schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ren%2C+W">Weijie Ren</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wenjia Xie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Ye Zhang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+H">Hang Yu</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Z">Zhengyu Tian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.03428v1-abstract-short" style="display: inline;"> The shock instability problem commonly arises in flow simulations involving strong shocks, particularly when employing high-order schemes, limiting their applications in hypersonic flow simulations. This study focuses on exploring the numerical characteristics and underlying mechanisms of shock instabilities in fifth-order finite-volume WENO schemes. To this end, for the first time, we have establ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.03428v1-abstract-full').style.display = 'inline'; document.getElementById('2308.03428v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.03428v1-abstract-full" style="display: none;"> The shock instability problem commonly arises in flow simulations involving strong shocks, particularly when employing high-order schemes, limiting their applications in hypersonic flow simulations. This study focuses on exploring the numerical characteristics and underlying mechanisms of shock instabilities in fifth-order finite-volume WENO schemes. To this end, for the first time, we have established the matrix stability analysis method for the fifth-order scheme. By predicting the evolution of perturbation errors in the exponential growth stage, this method provides quantitative insights into the behavior of shock-capturing and helps elucidate the mechanisms that cause shock instabilities. Results reveal that even dissipative solvers also suffer from shock instabilities when the spatial accuracy is increased to fifth-order. Further investigation indicates that this is due to the excessively high spatial accuracy of the WENO scheme near the numerical shock structure. Moreover, the shock instability problem of fifth-order schemes is demonstrated to be a multidimensional coupling problem. To stably capture strong shocks, it is crucial to have sufficient dissipation on transverse faces and ensure at least two points within the numerical shock structure in the direction perpendicular to the shock. The source location of instability is also clarified by the matrix stability analysis method, revealing that the instability arises from the numerical shock structure. Additionally, stability analysis demonstrates that local characteristic decomposition helps mitigate shock instabilities in high-order schemes, although the instability still persists. These conclusions pave the way for a better understanding of the shock instability in fifth-order schemes and provide guidance for the development of more reliable high-order shock-capturing methods for compressible flows with high Mach numbers. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.03428v1-abstract-full').style.display = 'none'; document.getElementById('2308.03428v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">42 pages, 28 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.07638">arXiv:2305.07638</a> <span> [<a href="https://arxiv.org/pdf/2305.07638">pdf</a>, <a href="https://arxiv.org/ps/2305.07638">ps</a>, <a href="https://arxiv.org/format/2305.07638">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> On the Partial Convexification for Low-Rank Spectral Optimization: Rank Bounds and Algorithms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.07638v2-abstract-short" style="display: inline;"> A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective subject to multiple two-sided linear matrix inequalities intersected with a low-rank and spectral constrained domain set. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain set by its convex hull) termed "LSOP-R," is often tractable and yields a high-quality solution. Th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.07638v2-abstract-full').style.display = 'inline'; document.getElementById('2305.07638v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.07638v2-abstract-full" style="display: none;"> A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective subject to multiple two-sided linear matrix inequalities intersected with a low-rank and spectral constrained domain set. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain set by its convex hull) termed "LSOP-R," is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of the feasible set of LSOP-R and prove their tightness for the domain sets with different matrix spaces. The proposed rank bounds recover two well-known results in the literature from a fresh angle and also allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to the original LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle, coupled with a rank-reduction algorithm, which ensures the output solution satisfies the theoretical rank bound. Finally, we numerically verify the strength of the LSOP-R and the efficacy of the proposed algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.07638v2-abstract-full').style.display = 'none'; document.getElementById('2305.07638v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.03281">arXiv:2305.03281</a> <span> [<a href="https://arxiv.org/pdf/2305.03281">pdf</a>, <a href="https://arxiv.org/format/2305.03281">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Fluid Dynamics">physics.flu-dyn</span> </div> </div> <p class="title is-5 mathjax"> Numerical stability analysis of shock-capturing methods for strong shocks I: second-order MUSCL schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ren%2C+W">Weijie Ren</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wenjia Xie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Ye Zhang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+H">Hang Yu</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Z">Zhengyu Tian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.03281v1-abstract-short" style="display: inline;"> Modern shock-capturing schemes often suffer from numerical shock anomalies if the flow field contains strong shocks, which may limit their further application in hypersonic flow computations. In the current study, we devote our efforts to exploring the primary numerical characteristics and the underlying mechanism of shock instability for second-order finite-volume schemes. To this end, we, for th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.03281v1-abstract-full').style.display = 'inline'; document.getElementById('2305.03281v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.03281v1-abstract-full" style="display: none;"> Modern shock-capturing schemes often suffer from numerical shock anomalies if the flow field contains strong shocks, which may limit their further application in hypersonic flow computations. In the current study, we devote our efforts to exploring the primary numerical characteristics and the underlying mechanism of shock instability for second-order finite-volume schemes. To this end, we, for the first time, develop the matrix stability analysis method for the finite-volume MUSCL approach. Such a linearized analysis method allows to investigate the shock instability problem of the finite-volume shock-capturing schemes in a quantitative and efficient manner. Results of the stability analysis demonstrate that the shock stability of second-order scheme is strongly related to the Riemann solver, Mach number, limiter function, numerical shock structure, and computational grid. Unique stability characteristics associated with these factors for second-order methods are revealed quantitatively with the established method. Source location of instability is also clarified by the matrix stability analysis method. Results show that the shock instability originates from the numerical shock structure. Such conclusions pave the way to better understand the shock instability problem and may shed new light on developing more reliable shock-capturing methods for compressible flows with high Mach number. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.03281v1-abstract-full').style.display = 'none'; document.getElementById('2305.03281v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">43 pages,23 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.01737">arXiv:2302.01737</a> <span> [<a href="https://arxiv.org/pdf/2302.01737">pdf</a>, <a href="https://arxiv.org/ps/2302.01737">ps</a>, <a href="https://arxiv.org/format/2302.01737">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> ALSO-X#: Better Convex Approximations for Distributionally Robust Chance Constrained Programs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+N">Nan Jiang</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.01737v1-abstract-short" style="display: inline;"> This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recentl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.01737v1-abstract-full').style.display = 'inline'; document.getElementById('2302.01737v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.01737v1-abstract-full" style="display: none;"> This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recently, ALSO-X has been proven to outperform the conditional value-at-risk (CVaR) approximation of a regular chance constrained program when the deterministic set is convex. In this work, we relax this assumption by introducing a new ALSO-X\# method for solving DRCCPs. Namely, in the bilevel reformulations of ALSO-X and CVaR approximation, we observe that the lower-level ALSO-X is a special case of the lower-level CVaR approximation and the upper-level CVaR approximation is more restricted than the one in ALSO-X. This observation motivates us to propose the ALSO-X\#, which still resembles a bilevel formulation -- in the lower-level problem, we adopt the more general CVaR approximation, and for the upper-level one, we choose the less restricted ALSO-X. We show that ALSO-X\# can always be better than the CVaR approximation and can outperform ALSO-X under regular chance constrained programs and type $\infty-$Wasserstein ambiguity set. We also provide new sufficient conditions under which ALSO-X\# outputs an optimal solution to a DRCCP. We apply the proposed ALSO-X\# to a wireless communication problem and numerically demonstrate that the solution quality can be even better than the exact method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.01737v1-abstract-full').style.display = 'none'; document.getElementById('2302.01737v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.14704">arXiv:2211.14704</a> <span> [<a href="https://arxiv.org/pdf/2211.14704">pdf</a>, <a href="https://arxiv.org/ps/2211.14704">ps</a>, <a href="https://arxiv.org/format/2211.14704">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Quantum State Transfer in Graphs with Tails </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bernard%2C+P">Pierre-Antoine Bernard</a>, <a href="/search/math?searchtype=author&query=Tamon%2C+C">Christino Tamon</a>, <a href="/search/math?searchtype=author&query=Vinet%2C+L">Luc Vinet</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weichen Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2211.14704v1-abstract-short" style="display: inline;"> We consider quantum state transfer on finite graphs which are attached to infinite paths. The finite graph represents an operational quantum system for performing useful quantum information tasks. In contrast, the infinite paths represent external infinite-dimensional systems which have limited (but nontrivial) interaction with the finite quantum system. We show that {\em perfect} state transfer c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.14704v1-abstract-full').style.display = 'inline'; document.getElementById('2211.14704v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.14704v1-abstract-full" style="display: none;"> We consider quantum state transfer on finite graphs which are attached to infinite paths. The finite graph represents an operational quantum system for performing useful quantum information tasks. In contrast, the infinite paths represent external infinite-dimensional systems which have limited (but nontrivial) interaction with the finite quantum system. We show that {\em perfect} state transfer can surprisingly still occur on the finite graph even in the presence of the infinite tails. Our techniques are based on a decoupling theorem for eventually-free Jacobi matrices, equitable partitions, and standard Lie theoretic arguments. Through these methods, we rehabilitate the notion of a dark subspace which had been so far viewed in an unflattering light. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.14704v1-abstract-full').style.display = 'none'; document.getElementById('2211.14704v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.16191">arXiv:2210.16191</a> <span> [<a href="https://arxiv.org/pdf/2210.16191">pdf</a>, <a href="https://arxiv.org/format/2210.16191">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> On the Exactness of Dantzig-Wolfe Relaxation for Rank Constrained Optimization Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.16191v3-abstract-short" style="display: inline;"> In the rank-constrained optimization problem (RCOP), it minimizes a linear objective function over a prespecified closed rank-constrained domain set and $m$ generic two-sided linear matrix inequalities. Motivated by the Dantzig-Wolfe (DW) decomposition, a popular approach of solving many nonconvex optimization problems, we investigate the strength of DW relaxation (DWR) of the RCOP, which admits t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.16191v3-abstract-full').style.display = 'inline'; document.getElementById('2210.16191v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.16191v3-abstract-full" style="display: none;"> In the rank-constrained optimization problem (RCOP), it minimizes a linear objective function over a prespecified closed rank-constrained domain set and $m$ generic two-sided linear matrix inequalities. Motivated by the Dantzig-Wolfe (DW) decomposition, a popular approach of solving many nonconvex optimization problems, we investigate the strength of DW relaxation (DWR) of the RCOP, which admits the same formulation as RCOP except replacing the domain set by its closed convex hull. Notably, our goal is to characterize conditions under which the DWR matches RCOP for any m two-sided linear matrix inequalities. From the primal perspective, we develop the first-known simultaneously necessary and sufficient conditions that achieve: (i) extreme point exactness -- all the extreme points of the DWR feasible set belong to that of the RCOP; (ii) convex hull exactness -- the DWR feasible set is identical to the closed convex hull of RCOP feasible set; and (iii) objective exactness -- the optimal values of the DWR and RCOP coincide. The proposed conditions unify, refine, and extend the existing exactness results in the quadratically constrained quadratic program (QCQP) and fair unsupervised learning. These conditions can be very useful to identify new results, including the extreme point exactness for a QCQP problem that admits an inhomogeneous objective function with two homogeneous two-sided quadratic constraints and the convex hull exactness for fair SVD. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.16191v3-abstract-full').style.display = 'none'; document.getElementById('2210.16191v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.01433">arXiv:2209.01433</a> <span> [<a href="https://arxiv.org/pdf/2209.01433">pdf</a>, <a href="https://arxiv.org/format/2209.01433">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gomez%2C+A">Andres Gomez</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.01433v1-abstract-short" style="display: inline;"> In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we characterize the structu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.01433v1-abstract-full').style.display = 'inline'; document.getElementById('2209.01433v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.01433v1-abstract-full" style="display: none;"> In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we characterize the structure of the convex hull, and show that it can be formally studied using polyhedral theory. Moreover, we show that although perspective relaxation in the literature for this set fails to match the structure of its convex hull, it is guaranteed to be a close approximation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.01433v1-abstract-full').style.display = 'none'; document.getElementById('2209.01433v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.03589">arXiv:2208.03589</a> <span> [<a href="https://arxiv.org/pdf/2208.03589">pdf</a>, <a href="https://arxiv.org/format/2208.03589">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> D-optimal Data Fusion: Exact and Approximation Algorithms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Fampa%2C+M">Marcia Fampa</a>, <a href="/search/math?searchtype=author&query=Lee%2C+J">Jon Lee</a>, <a href="/search/math?searchtype=author&query=Qiu%2C+F">Feng Qiu</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a>, <a href="/search/math?searchtype=author&query=Yao%2C+R">Rui Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.03589v1-abstract-short" style="display: inline;"> We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P $=$ NP. Therefore, to solve the DDF problem effectively, w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.03589v1-abstract-full').style.display = 'inline'; document.getElementById('2208.03589v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.03589v1-abstract-full" style="display: none;"> We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P $=$ NP. Therefore, to solve the DDF problem effectively, we propose two convex integer-programming formulations and investigate their corresponding complementary and Lagrangian-dual problems. We also develop scalable randomized-sampling and local-search algorithms with provable performance guarantees. Leveraging the concavity of the objective functions in the two proposed formulations, we design an exact algorithm, aimed at solving the DDF problem to optimality. We further derive a family of submodular valid inequalities and optimality cuts, which can significantly enhance the algorithm performance. Finally, we test our algorithms using real-world data on the new phasor-measurement-units placement problem for modern power grids, considering the existing conventional sensors. Our numerical study demonstrates the efficiency of our exact algorithm and the scalability and high-quality outputs of our approximation algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.03589v1-abstract-full').style.display = 'none'; document.getElementById('2208.03589v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2204.04355">arXiv:2204.04355</a> <span> [<a href="https://arxiv.org/pdf/2204.04355">pdf</a>, <a href="https://arxiv.org/format/2204.04355">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Of Shadows and Gaps in Spatial Search </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chan%2C+A">Ada Chan</a>, <a href="/search/math?searchtype=author&query=Godsil%2C+C">Chris Godsil</a>, <a href="/search/math?searchtype=author&query=Tamon%2C+C">Christino Tamon</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weichen Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2204.04355v2-abstract-short" style="display: inline;"> Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.04355v2-abstract-full').style.display = 'inline'; document.getElementById('2204.04355v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2204.04355v2-abstract-full" style="display: none;"> Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty et al. (Physical Review A, 102:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.04355v2-abstract-full').style.display = 'none'; document.getElementById('2204.04355v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, 3 figures. Current version: fixed minor typo in Theorem 1 and proof for cycles; clarified assumption on model</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Quantum Information and Computation, 22(13&14):1110-1131, 2022 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.13924">arXiv:2109.13924</a> <span> [<a href="https://arxiv.org/pdf/2109.13924">pdf</a>, <a href="https://arxiv.org/ps/2109.13924">ps</a>, <a href="https://arxiv.org/format/2109.13924">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^n$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gui%2C+C">Changfeng Gui</a>, <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yeyao Hu</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weihong Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.13924v1-abstract-short" style="display: inline;"> In this article we present various uniqueness and existence results for Q-curvature type equations with a Paneitz operator on $\s^n$ in axially symmetric function spaces. In particular, we show uniqueness results for $n=6, 8$ and improve the best constant of Beckner's inequality in these dimensions for axially symmetric functions under the constraint that their centers of mass are at the origin. A… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.13924v1-abstract-full').style.display = 'inline'; document.getElementById('2109.13924v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.13924v1-abstract-full" style="display: none;"> In this article we present various uniqueness and existence results for Q-curvature type equations with a Paneitz operator on $\s^n$ in axially symmetric function spaces. In particular, we show uniqueness results for $n=6, 8$ and improve the best constant of Beckner's inequality in these dimensions for axially symmetric functions under the constraint that their centers of mass are at the origin. As a consequence, the associated first Szeg枚 limit theorem is also proven for axially symmetric functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.13924v1-abstract-full').style.display = 'none'; document.getElementById('2109.13924v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: substantial text overlap with arXiv:2109.13390</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.13390">arXiv:2109.13390</a> <span> [<a href="https://arxiv.org/pdf/2109.13390">pdf</a>, <a href="https://arxiv.org/ps/2109.13390">ps</a>, <a href="https://arxiv.org/format/2109.13390">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gui%2C+C">Changfeng Gui</a>, <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yeyao Hu</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weihong Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.13390v1-abstract-short" style="display: inline;"> We show that axially symmetric solutions on $\mathbb{S}^4$ to a constant $Q$-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter $伪$ in front of the Paneitz operator belongs to $[\frac{473 + \sqrt{209329}}{1800}\approx0.517, 1)$. This is in contrast to the case $伪=1$, where a family of solutions exist, known as standard bub… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.13390v1-abstract-full').style.display = 'inline'; document.getElementById('2109.13390v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.13390v1-abstract-full" style="display: none;"> We show that axially symmetric solutions on $\mathbb{S}^4$ to a constant $Q$-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter $伪$ in front of the Paneitz operator belongs to $[\frac{473 + \sqrt{209329}}{1800}\approx0.517, 1)$. This is in contrast to the case $伪=1$, where a family of solutions exist, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on $ \mathbb{S}^2$. As a consequence, we prove an improved Beckner's inequality on $\mathbb{S}^4$ for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when $伪=\frac15$ by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for $伪\in (\frac15, \frac12)$ via a bifurcation method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.13390v1-abstract-full').style.display = 'none'; document.getElementById('2109.13390v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.09123">arXiv:2106.09123</a> <span> [<a href="https://arxiv.org/pdf/2106.09123">pdf</a>, <a href="https://arxiv.org/format/2106.09123">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Second-Order Conic and Polyhedral Approximations of the Exponential Cone: Application to Mixed-Integer Exponential Conic Programs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+Q">Qing Ye</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.09123v3-abstract-short" style="display: inline;"> Exponents and logarithms are fundamental components in many important applications such as logistic regression, maximum likelihood, relative entropy, and so on. Since the exponential cone can be viewed as the epigraph of perspective of the natural exponential function or the hypograph of perspective of the natural logarithm function, many mixed-integer convex programs involving exponential or loga… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.09123v3-abstract-full').style.display = 'inline'; document.getElementById('2106.09123v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.09123v3-abstract-full" style="display: none;"> Exponents and logarithms are fundamental components in many important applications such as logistic regression, maximum likelihood, relative entropy, and so on. Since the exponential cone can be viewed as the epigraph of perspective of the natural exponential function or the hypograph of perspective of the natural logarithm function, many mixed-integer convex programs involving exponential or logarithm functions can be recast as mixed-integer exponential conic programs (MIECPs). However, unlike mixed-integer linear programs (MILPs) and mixed-integer second-order conic programs (MISOCPs), MIECPs are still under development. To harvest the past efforts on MILPs and MISOCPs, this paper presents second-order conic (SOC) and polyhedral approximation schemes for the exponential cone with application to MIECPs. To do so, we first extend and generalize existing SOC approximation approaches in the extended space, propose new scaling and shifting methods, prove approximation accuracies, and derive lower bounds of approximations. We then study the polyhedral outer approximation of the exponential cones in the original space using gradient inequalities, show its approximation accuracy, and derive a lower bound of the approximation. When implementing SOC approximations, we suggest learning the approximation pattern by testing smaller cases and then applying it to the large-scale ones; and for the polyhedral approximation, we suggest using the branch and cut method for MIECPs. Our numerical study shows that the proposed methods show speed-ups over solver MOSEK for MIECPs, and the scaling, shifting, and polyhedral outer approximation methods work very well. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.09123v3-abstract-full').style.display = 'none'; document.getElementById('2106.09123v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages, 9 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.04288">arXiv:2106.04288</a> <span> [<a href="https://arxiv.org/pdf/2106.04288">pdf</a>, <a href="https://arxiv.org/ps/2106.04288">ps</a>, <a href="https://arxiv.org/format/2106.04288">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219199723500086">10.1142/S0219199723500086 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Infinitely many solutions for Schr枚dinger-Newton equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yeyao Hu</a>, <a href="/search/math?searchtype=author&query=Jevnikar%2C+A">Aleks Jevnikar</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weihong Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.04288v1-abstract-short" style="display: inline;"> We prove the existence of infinitely many non-radial positive solutions for the Schr枚dinger-Newton system $$ \left\{\begin{array}{ll} 螖u- V(|x|)u + 唯u=0, &x\in\mathbb{R}^3,\newline 螖唯+\frac12 u^2=0, &x\in\mathbb{R}^3, \end{array}\right. $$ provided that $V(r)$ has the following behavior at infinity:… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.04288v1-abstract-full').style.display = 'inline'; document.getElementById('2106.04288v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.04288v1-abstract-full" style="display: none;"> We prove the existence of infinitely many non-radial positive solutions for the Schr枚dinger-Newton system $$ \left\{\begin{array}{ll} 螖u- V(|x|)u + 唯u=0, &x\in\mathbb{R}^3,\newline 螖唯+\frac12 u^2=0, &x\in\mathbb{R}^3, \end{array}\right. $$ provided that $V(r)$ has the following behavior at infinity: $$ V(r)=V_0+\frac{a}{r^m}+O\left(\frac{1}{r^{m+胃}}\right) \quad\mbox{ as } r\rightarrow\infty, $$ where $\frac12\le m<1$ and $a, V_0, 胃$ are some positive constants. In particular, for any $s$ large we use a reduction method to construct $s-$bump solutions lying on a circle of radius $r\sim (s\log s)^{\frac{1}{1-m}}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.04288v1-abstract-full').style.display = 'none'; document.getElementById('2106.04288v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35B40; 35B45; 35J40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> CCM 2023 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.02917">arXiv:2106.02917</a> <span> [<a href="https://arxiv.org/pdf/2106.02917">pdf</a>, <a href="https://arxiv.org/format/2106.02917">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Economics">econ.GN</span> </div> </div> <p class="title is-5 mathjax"> On the Stratification of Product Portfolios </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Govindan%2C+V">Vikram Govindan</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wei Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.02917v1-abstract-short" style="display: inline;"> Stratifying commercial product portfolios into multiple classes of decreasing priority, ABCD analysis, is a common supply chain tool. Key planning parameters that drive strategic and execution priorities are tied to the resulting segmentation. These priorities in turn drive supply chain performance. For large product assortments, manual segmentation is infeasible so an automated algorithm is neede… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.02917v1-abstract-full').style.display = 'inline'; document.getElementById('2106.02917v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.02917v1-abstract-full" style="display: none;"> Stratifying commercial product portfolios into multiple classes of decreasing priority, ABCD analysis, is a common supply chain tool. Key planning parameters that drive strategic and execution priorities are tied to the resulting segmentation. These priorities in turn drive supply chain performance. For large product assortments, manual segmentation is infeasible so an automated algorithm is needed. We therefore advocate that careful attention be paid to the design of such an ABCD algorithm and present three key features that can be incorporated into such a calculation to improve its quality and commercial utility. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.02917v1-abstract-full').style.display = 'none'; document.getElementById('2106.02917v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 5 tables, 5 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.03179">arXiv:2105.03179</a> <span> [<a href="https://arxiv.org/pdf/2105.03179">pdf</a>, <a href="https://arxiv.org/format/2105.03179">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Beyond Symmetry: Best Submatrix Selection for the Sparse Truncated SVD </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.03179v3-abstract-short" style="display: inline;"> Truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation, has been successfully applied to many domains such as biology, healthcare, and others, where high-dimensional datasets are prevalent. To enhance the interpretability of the truncated SVD, sparse SVD (SSVD) is introduced to select a few rows and columns of the original matrix along with the low rank… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.03179v3-abstract-full').style.display = 'inline'; document.getElementById('2105.03179v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.03179v3-abstract-full" style="display: none;"> Truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation, has been successfully applied to many domains such as biology, healthcare, and others, where high-dimensional datasets are prevalent. To enhance the interpretability of the truncated SVD, sparse SVD (SSVD) is introduced to select a few rows and columns of the original matrix along with the low rank approximation. Different from the literature, this paper presents a novel SSVD formulation that can select the best submatrix precisely up to a given size to maximize its truncated Ky Fan norm. The fact that the SSVD problem is NP-hard motivates us to study effective algorithms with provable performance guarantees. To do so, we first reformulate SSVD as a mixed-integer semidefinite program, which can be solved exactly for small- or medium-sized instances by a customized branch and cut algorithm with closed-form cuts, and is extremely useful to evaluate the quality of approximation algorithms. We next develop three selection algorithms based on different selection criteria and two searching algorithms -- greedy and local search. We prove the approximation ratios for all the approximation algorithms and show that all the ratios are tight, i.e., we demonstrate that these approximation ratios are unimprovable. Finally, our numerical study demonstrates the high solution quality and computational efficiency of the proposed algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.03179v3-abstract-full').style.display = 'none'; document.getElementById('2105.03179v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.10809">arXiv:2103.10809</a> <span> [<a href="https://arxiv.org/pdf/2103.10809">pdf</a>, <a href="https://arxiv.org/format/2103.10809">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Generalized fractional grey system models: Memory effects perspective </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wanli Xie</a>, <a href="/search/math?searchtype=author&query=Wu%2C+W">Wen-Ze Wu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Chong Liu</a>, <a href="/search/math?searchtype=author&query=Goh%2C+M">Mark Goh</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.10809v2-abstract-short" style="display: inline;"> As an essential characteristics of fractional calculus, the memory effect is served as one of key factors to deal with diverse practical issues, thus has been received extensive attention since it was born. By combining the fractional derivative with memory effects and grey modeling theory, this paper aims to construct an unified framework for the commonly-used fractional grey models already in pl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.10809v2-abstract-full').style.display = 'inline'; document.getElementById('2103.10809v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.10809v2-abstract-full" style="display: none;"> As an essential characteristics of fractional calculus, the memory effect is served as one of key factors to deal with diverse practical issues, thus has been received extensive attention since it was born. By combining the fractional derivative with memory effects and grey modeling theory, this paper aims to construct an unified framework for the commonly-used fractional grey models already in place. In particular, by taking different kernel and normalization functions, this framework can deduce some other new fractional grey models. To further improve the prediction performance, the four popular intelligent algorithms are employed to determine the emerging coefficients for the UFGM(1,1) model. Two published cases are then utilized to verify the validity of the UFGM(1,1) model and explore the effects of fractional accumulation order and initial value on the prediction accuracy, respectively. Finally, this model is also applied to dealing with two real examples so as to further explain its efficacy and equally show how to use the unified framework in practical applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.10809v2-abstract-full').style.display = 'none'; document.getElementById('2103.10809v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.06374">arXiv:2103.06374</a> <span> [<a href="https://arxiv.org/pdf/2103.06374">pdf</a>, <a href="https://arxiv.org/format/2103.06374">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Engineering, Finance, and Science">cs.CE</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> </div> </div> <p class="title is-5 mathjax"> A unified construction of all-speed HLL-type schemes for hypersonic heating computations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wenjia Xie</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Z">Zhengyu Tian</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Ye Zhang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+H">Hang Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.06374v1-abstract-short" style="display: inline;"> In this paper, a unified framework to develop all-speed HLL-type schemes for hypersonic heating computations is constructed. Such a unified construction method combines two effective improving techniques: a shock robustness improvement and a low-Mach number fix. It is implemented by properly modifying the approximate solutions of the local Riemann problem in the HLL framework, resulting in two all… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.06374v1-abstract-full').style.display = 'inline'; document.getElementById('2103.06374v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.06374v1-abstract-full" style="display: none;"> In this paper, a unified framework to develop all-speed HLL-type schemes for hypersonic heating computations is constructed. Such a unified construction method combines two effective improving techniques: a shock robustness improvement and a low-Mach number fix. It is implemented by properly modifying the approximate solutions of the local Riemann problem in the HLL framework, resulting in two all-speed HLL-type schemes, namely ASHLLC and ASHLLEM solvers. Results from both numerical analysis and experiments demonstrate that the newly proposed schemes not only preserve desirable properties of their original versions, but are also able to provide accurate and robust solutions for complex flows ranging from low-Mach number incompressible to hypersonic compressible regimes. Thus, both the ASHLLC and ASHLLEM schemes can be used as reliable methods for hypersonic heating computations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.06374v1-abstract-full').style.display = 'none'; document.getElementById('2103.06374v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.04763">arXiv:2012.04763</a> <span> [<a href="https://arxiv.org/pdf/2012.04763">pdf</a>, <a href="https://arxiv.org/format/2012.04763">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> ALSO-X and ALSO-X+: Better Convex Approximations for Chance Constrained Programs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+N">Nan Jiang</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2012.04763v4-abstract-short" style="display: inline;"> In a chance constrained program (CCP), the decision-makers aim to seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk (CVaR) has been know… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.04763v4-abstract-full').style.display = 'inline'; document.getElementById('2012.04763v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.04763v4-abstract-full" style="display: none;"> In a chance constrained program (CCP), the decision-makers aim to seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk (CVaR) has been known to be the best for more than a decade. This paper studies and generalizes the ALSO-X, originally proposed by Ahmed, Luedtke, SOng, and Xie (2017), for solving a CCP. We first show that the ALSO-X resembles a bilevel optimization, where the upper-level problem is to find the best objective function value and enforce the feasibility of a CCP for a given decision from the lower-level problem, and the lower-level problem is to minimize the expectation of constraint violations subject to the upper bound of the objective function value provided by the upper-level problem. This interpretation motivates us to prove that when uncertain constraints are convex in the decision variables, ALSO-X always outperforms the CVaR approximation. We further show (i) sufficient conditions under which ALSO-X can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance ALSO-X with a convergent alternating minimization method (ALSO-X+); (iii) extensions of ALSO-X and ALSO-X+ to solve distributionally robust chance constrained programs (DRCCPs) under $\infty$-Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.04763v4-abstract-full').style.display = 'none'; document.getElementById('2012.04763v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">60 pages, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.12438">arXiv:2008.12438</a> <span> [<a href="https://arxiv.org/pdf/2008.12438">pdf</a>, <a href="https://arxiv.org/format/2008.12438">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Exact and Approximation Algorithms for Sparse PCA </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.12438v1-abstract-short" style="display: inline;"> Sparse PCA (SPCA) is a fundamental model in machine learning and data analytics, which has witnessed a variety of application areas such as finance, manufacturing, biology, healthcare. To select a prespecified-size principal submatrix from a covariance matrix to maximize its largest eigenvalue for the better interpretability purpose, SPCA advances the conventional PCA with both feature selection a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.12438v1-abstract-full').style.display = 'inline'; document.getElementById('2008.12438v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.12438v1-abstract-full" style="display: none;"> Sparse PCA (SPCA) is a fundamental model in machine learning and data analytics, which has witnessed a variety of application areas such as finance, manufacturing, biology, healthcare. To select a prespecified-size principal submatrix from a covariance matrix to maximize its largest eigenvalue for the better interpretability purpose, SPCA advances the conventional PCA with both feature selection and dimensionality reduction. This paper proposes two exact mixed-integer SDPs (MISDPs) by exploiting the spectral decomposition of the covariance matrix and the properties of the largest eigenvalues. We then analyze the theoretical optimality gaps of their continuous relaxation values and prove that they are stronger than that of the state-of-art one. We further show that the continuous relaxations of two MISDPs can be recast as saddle point problems without involving semi-definite cones, and thus can be effectively solved by first-order methods such as the subgradient method. Since off-the-shelf solvers, in general, have difficulty in solving MISDPs, we approximate SPCA with arbitrary accuracy by a mixed-integer linear program (MILP) of a similar size as MISDPs. To be more scalable, we also analyze greedy and local search algorithms, prove their first-known approximation ratios, and show that the approximation ratios are tight. Our numerical study demonstrates that the continuous relaxation values of the proposed MISDPs are quite close to optimality, the proposed MILP model can solve small and medium-size instances to optimality, and the approximation algorithms work very well for all the instances. Finally, we extend the analyses to Rank-one Sparse SVD (R1-SSVD) with non-symmetric matrices and Sparse Fair PCA (SFPCA) when there are multiple covariance matrices, each corresponding to a protected group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.12438v1-abstract-full').style.display = 'none'; document.getElementById('2008.12438v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">49 pages, 1 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.00803">arXiv:2008.00803</a> <span> [<a href="https://arxiv.org/pdf/2008.00803">pdf</a>, <a href="https://arxiv.org/ps/2008.00803">ps</a>, <a href="https://arxiv.org/format/2008.00803">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Mathematics">math.GM</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.chaos.2020.110285">10.1016/j.chaos.2020.110285 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Continuous grey model with conformable fractional derivative </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wanli Xie</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Caixia Liu</a>, <a href="/search/math?searchtype=author&query=Li%2C+W">Weidong Li</a>, <a href="/search/math?searchtype=author&query=Wu%2C+W">Wenze Wu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Chong Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.00803v2-abstract-short" style="display: inline;"> The existing fractional grey prediction models mainly use discrete fractional-order difference and accumulation, but in the actual modeling, continuous fractional-order calculus has been proved to have many excellent properties, such as hereditary. Now there are grey models established with continuous fractional-order calculus method, and they have achieved good results. However, the models are ve… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.00803v2-abstract-full').style.display = 'inline'; document.getElementById('2008.00803v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.00803v2-abstract-full" style="display: none;"> The existing fractional grey prediction models mainly use discrete fractional-order difference and accumulation, but in the actual modeling, continuous fractional-order calculus has been proved to have many excellent properties, such as hereditary. Now there are grey models established with continuous fractional-order calculus method, and they have achieved good results. However, the models are very complicated in the calculation and are not conducive to the actual application. In order to further simplify and improve the grey prediction models with continuous fractional-order derivative, we propose a simple and effective grey model based on conformable fractional derivatives in this paper, and two practical cases are used to demonstrate the validity of the proposed model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.00803v2-abstract-full').style.display = 'none'; document.getElementById('2008.00803v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2003.00630">arXiv:2003.00630</a> <span> [<a href="https://arxiv.org/pdf/2003.00630">pdf</a>, <a href="https://arxiv.org/format/2003.00630">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Distributionally Robust Bottleneck Combinatorial Problems: Uncertainty Quantification and Robust Decision Making </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jie Zhang</a>, <a href="/search/math?searchtype=author&query=Ahmed%2C+S">Shabbir Ahmed</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2003.00630v3-abstract-short" style="display: inline;"> This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.00630v3-abstract-full').style.display = 'inline'; document.getElementById('2003.00630v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2003.00630v3-abstract-full" style="display: none;"> This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max-min-max form. Fortunately, the alternative forms of the bottleneck combinatorial problems from their blockers allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-and-passenger matching that minimizes the unfairness. This gives rise to the decision-making DRBCP (denoted by DRBCP-D). For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixed-integer program. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.00630v3-abstract-full').style.display = 'none'; document.getElementById('2003.00630v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">32 pages, 4 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.08537">arXiv:2001.08537</a> <span> [<a href="https://arxiv.org/pdf/2001.08537">pdf</a>, <a href="https://arxiv.org/format/2001.08537">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Best Principal Submatrix Selection for the Maximum Entropy Sampling Problem: Scalable Algorithms and Performance Guarantees </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongchun Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.08537v3-abstract-short" style="display: inline;"> This paper studies a classic maximum entropy sampling problem (MESP), which aims to select the most informative principal submatrix of a prespecified size from a covariance matrix. MESP has been widely applied to many areas, including healthcare, power system, manufacturing and data science. By investigating its Lagrangian dual and primal characterization, we derive a novel convex integer program… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.08537v3-abstract-full').style.display = 'inline'; document.getElementById('2001.08537v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.08537v3-abstract-full" style="display: none;"> This paper studies a classic maximum entropy sampling problem (MESP), which aims to select the most informative principal submatrix of a prespecified size from a covariance matrix. MESP has been widely applied to many areas, including healthcare, power system, manufacturing and data science. By investigating its Lagrangian dual and primal characterization, we derive a novel convex integer program for MESP and show that its continuous relaxation yields a near-optimal solution. The results motivate us to study an efficient sampling algorithm and develop its approximation bound for MESP, which improves the best-known bound in literature. We then provide an efficient deterministic implementation of the sampling algorithm with the same approximation bound. By developing new mathematical tools for the singular matrices and analyzing the Lagrangian dual of the proposed convex integer program, we investigate the widely-used local search algorithm and prove its first-known approximation bound for MESP. The proof techniques further inspire us with an efficient implementation of the local search algorithm. Our numerical experiments demonstrate that these approximation algorithms can efficiently solve medium-sized and large-scale instances to near-optimality. Our proposed algorithms are coded and released as open-source software. Finally, we extend the analyses to the A-Optimal MESP (A-MESP), where the objective is to minimize the trace of the inverse of the selected principal submatrix. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.08537v3-abstract-full').style.display = 'none'; document.getElementById('2001.08537v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">62 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.01953">arXiv:1911.01953</a> <span> [<a href="https://arxiv.org/pdf/1911.01953">pdf</a>, <a href="https://arxiv.org/format/1911.01953">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A Note on Quantum Markov Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tamon%2C+C">Christino Tamon</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Weichen Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.01953v1-abstract-short" style="display: inline;"> The study of Markov models is central to control theory and machine learning. A quantum analogue of partially observable Markov decision process was studied in (Barry, Barry, and Aaronson, Phys. Rev. A, 90, 2014). It was proved that goal-state reachability is undecidable in the quantum setting, whereas it is decidable classically. In contrast to this classical-to-quantum transition from decidable… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.01953v1-abstract-full').style.display = 'inline'; document.getElementById('1911.01953v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.01953v1-abstract-full" style="display: none;"> The study of Markov models is central to control theory and machine learning. A quantum analogue of partially observable Markov decision process was studied in (Barry, Barry, and Aaronson, Phys. Rev. A, 90, 2014). It was proved that goal-state reachability is undecidable in the quantum setting, whereas it is decidable classically. In contrast to this classical-to-quantum transition from decidable to undecidable, we observe that the problem of approximating the optimal policy which maximizes the average discounted reward over an infinite horizon remains decidable in the quantum setting. Given that most relevant problems related to Markov decision process are undecidable classically (which immediately implies undecidability in the quantum case), this provides one of the few examples where the quantum problem is tractable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.01953v1-abstract-full').style.display = 'none'; document.getElementById('1911.01953v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 4 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.05863">arXiv:1910.05863</a> <span> [<a href="https://arxiv.org/pdf/1910.05863">pdf</a>, <a href="https://arxiv.org/format/1910.05863">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Global-Local Metamodel Assisted Two-Stage Optimization via Simulation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wei Xie</a>, <a href="/search/math?searchtype=author&query=Yi%2C+Y">Yuan Yi</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+H">Hua Zheng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1910.05863v1-abstract-short" style="display: inline;"> To integrate strategic, tactical and operational decisions, the two-stage optimization has been widely used to guide dynamic decision making. In this paper, we study the two-stage stochastic programming for complex systems with unknown response estimated by simulation. We introduce the global-local metamodel assisted two-stage optimization via simulation that can efficiently employ the simulation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.05863v1-abstract-full').style.display = 'inline'; document.getElementById('1910.05863v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.05863v1-abstract-full" style="display: none;"> To integrate strategic, tactical and operational decisions, the two-stage optimization has been widely used to guide dynamic decision making. In this paper, we study the two-stage stochastic programming for complex systems with unknown response estimated by simulation. We introduce the global-local metamodel assisted two-stage optimization via simulation that can efficiently employ the simulation resource to iteratively solve for the optimal first- and second-stage decisions. Specifically, at each visited first-stage decision, we develop a local metamodel to simultaneously solve a set of scenario-based second-stage optimization problems, which also allows us to estimate the optimality gap. Then, we construct a global metamodel accounting for the errors induced by: (1) using a finite number of scenarios to approximate the expected future cost occurring in the planning horizon, (2) second-stage optimality gap, and (3) finite visited first-stage decisions. Assisted by the global-local metamodel, we propose a new simulation optimization approach that can efficiently and iteratively search for the optimal first- and second-stage decisions. Our framework can guarantee the convergence of optimal solution for the discrete two-stage optimization with unknown objective, and the empirical study indicates that it achieves substantial efficiency and accuracy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.05863v1-abstract-full').style.display = 'none'; document.getElementById('1910.05863v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1908.08454">arXiv:1908.08454</a> <span> [<a href="https://arxiv.org/pdf/1908.08454">pdf</a>, <a href="https://arxiv.org/ps/1908.08454">ps</a>, <a href="https://arxiv.org/format/1908.08454">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Tractable Reformulations of Distributionally Robust Two-stage Stochastic Programs with $\infty-$Wasserstein Distance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1908.08454v1-abstract-short" style="display: inline;"> In the optimization under uncertainty, decision-makers first select a wait-and-see policy before any realization of uncertainty and then place a here-and-now decision after the uncertainty has been observed. Two-stage stochastic programming is a popular modeling paradigm for the optimization under uncertainty that the decision-makers first specifies a probability distribution, and then seek the be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.08454v1-abstract-full').style.display = 'inline'; document.getElementById('1908.08454v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1908.08454v1-abstract-full" style="display: none;"> In the optimization under uncertainty, decision-makers first select a wait-and-see policy before any realization of uncertainty and then place a here-and-now decision after the uncertainty has been observed. Two-stage stochastic programming is a popular modeling paradigm for the optimization under uncertainty that the decision-makers first specifies a probability distribution, and then seek the best decisions to jointly optimize the deterministic wait-and-see and expected here-and-now costs. In practice, such a probability distribution may not be fully available but is probably observable through an empirical dataset. Therefore, this paper studies distributionally robust two-stage stochastic program (DRTSP) which jointly optimizes the deterministic wait-and-see and worst-case expected here-and-now costs, and the probability distribution comes from a family of distributions which are centered at the empirical distribution using $\infty-$Wasserstein metric. There have been successful developments on deriving tractable approximations of the worst-case expected here-and-now cost in DRTSP. Unfortunately, limited results on exact tractable reformulations of DRTSP. This paper fills this gap by providing sufficient conditions under which the worst-case expected here-and-now cost in DRTSP can be efficiently computed via a tractable convex program. By exploring the properties of binary variables, the developed reformulation techniques are extended to DRTSP with binary random parameters. The main tractable reformulations in this paper are projected into the original decision space and thus can be interpreted as conventional two-stage stochastic programs under discrete support with extra penalty terms enforcing the robustness. These tractable results are further demonstrated to be sharp through complexity analysis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.08454v1-abstract-full').style.display = 'none'; document.getElementById('1908.08454v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.03794">arXiv:1811.03794</a> <span> [<a href="https://arxiv.org/pdf/1811.03794">pdf</a>, <a href="https://arxiv.org/ps/1811.03794">ps</a>, <a href="https://arxiv.org/format/1811.03794">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Physics and Society">physics.soc-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.cnsns.2019.01.012">10.1016/j.cnsns.2019.01.012 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Tetradic motif profiles of horizontal visibility graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wen-Jie Xie</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Rui-Qi Han</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+W">Wei-Xing Zhou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.03794v1-abstract-short" style="display: inline;"> Network motif analysis is a useful tool for the investigation of complex networks. We study the profiles of tetradic motifs in horizontal visibility graphs (HVGs) converted from multifractal binomial measures, fractional Gaussian noises, and heartbeat rates. The profiles of tetradic motifs contains the spatial information (visibility) and temporal information (relative magnitude) among the data po… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.03794v1-abstract-full').style.display = 'inline'; document.getElementById('1811.03794v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.03794v1-abstract-full" style="display: none;"> Network motif analysis is a useful tool for the investigation of complex networks. We study the profiles of tetradic motifs in horizontal visibility graphs (HVGs) converted from multifractal binomial measures, fractional Gaussian noises, and heartbeat rates. The profiles of tetradic motifs contains the spatial information (visibility) and temporal information (relative magnitude) among the data points in the corresponding time series. For multifractal binomial measures, the occurrence frequencies of the tetradic motifs are determined, which converge to a constant vector $(2/3,0,8/99,8/33,1/99,0)$. For fractional Gaussian noises, the motif occurrence frequencies are found to depend nonlinearly on the Hurst exponent and the length of time series. These findings suggest the potential ability of tetradic motif profiles in distinguishing different types of time series. Finally, we apply the tetradic motif analysis to heartbeat rates of healthy subjects, congestive heart failure (CHF) subjects, and atrial fibrillation (AF) subjects. Different subjects can be distinguished from the occurrence frequencies of tetradic motifs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.03794v1-abstract-full').style.display = 'none'; document.getElementById('1811.03794v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Commun Nonlinear Sci Numer Simulat 72 (2019) 544-551 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1809.02166">arXiv:1809.02166</a> <span> [<a href="https://arxiv.org/pdf/1809.02166">pdf</a>, <a href="https://arxiv.org/ps/1809.02166">ps</a>, <a href="https://arxiv.org/format/1809.02166">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Fine gradings and their Weyl groups for twisted Heisenberg Lie superalgebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Wenjuan Xie</a>, <a href="/search/math?searchtype=author&query=Liu%2C+W">Wende Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1809.02166v1-abstract-short" style="display: inline;"> In this paper we define the so-called twisted Heisenberg superalgebras over the complex number field by adding derivations to Heisenberg superalgebras. We classify the fine gradings up to equivalence on twisted Heisenberg superalgebras and determine the Weyl groups of those gradings. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1809.02166v1-abstract-full" style="display: none;"> In this paper we define the so-called twisted Heisenberg superalgebras over the complex number field by adding derivations to Heisenberg superalgebras. We classify the fine gradings up to equivalence on twisted Heisenberg superalgebras and determine the Weyl groups of those gradings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.02166v1-abstract-full').style.display = 'none'; document.getElementById('1809.02166v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages. arXiv admin note: text overlap with arXiv:1405.4093 by other authors</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B70; 17B40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1808.07953">arXiv:1808.07953</a> <span> [<a href="https://arxiv.org/pdf/1808.07953">pdf</a>, <a href="https://arxiv.org/format/1808.07953">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1742-5468/ab0c16">10.1088/1742-5468/ab0c16 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Thermal conductivity and local thermodynamic equilibrium of stochastic energy exchange models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yao Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wenbo Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1808.07953v2-abstract-short" style="display: inline;"> In this paper we study macroscopic thermodynamic properties of a stochastic microscopic heat conduction model that is reduced from deterministic problems. Our goal is to numerically check how the `low energy site effect' inherited from the deterministic model would affect the macroscopic thermodynamic properties such as the thermal conductivity and the local thermodynamic equilibrium. After a seri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.07953v2-abstract-full').style.display = 'inline'; document.getElementById('1808.07953v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.07953v2-abstract-full" style="display: none;"> In this paper we study macroscopic thermodynamic properties of a stochastic microscopic heat conduction model that is reduced from deterministic problems. Our goal is to numerically check how the `low energy site effect' inherited from the deterministic model would affect the macroscopic thermodynamic properties such as the thermal conductivity and the local thermodynamic equilibrium. After a series of numerical computations, our conclusion is that neither the thermal conductivity nor the existence of local thermodynamic equilibrium is qualitatively changed by this effect. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.07953v2-abstract-full').style.display = 'none'; document.getElementById('1808.07953v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Addressed reviewer's comments in V2</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82C05; 82C80; 60J22 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1806.07418">arXiv:1806.07418</a> <span> [<a href="https://arxiv.org/pdf/1806.07418">pdf</a>, <a href="https://arxiv.org/format/1806.07418">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> On Distributionally Robust Chance Constrained Programs with Wasserstein Distance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1806.07418v4-abstract-short" style="display: inline;"> This paper studies a distributionally robust chance constrained program (DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should be satisfied with a probability at least a given threshold for all the probability distributions of the uncertain parameters within a chosen Wasserstein distance from an empirical distribution. In this work, we investigate equivalent reformulations… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.07418v4-abstract-full').style.display = 'inline'; document.getElementById('1806.07418v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.07418v4-abstract-full" style="display: none;"> This paper studies a distributionally robust chance constrained program (DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should be satisfied with a probability at least a given threshold for all the probability distributions of the uncertain parameters within a chosen Wasserstein distance from an empirical distribution. In this work, we investigate equivalent reformulations and approximations of such problems. We first show that a DRCCP can be reformulated as a conditional value-at-risk constrained optimization problem, and thus admits tight inner and outer approximations. We also show that a DRCCP of bounded feasible region is mixed integer representable by introducing big-M coefficients and additional binary variables. For a DRCCP with pure binary decision variables, by exploring the submodular structure, we show that it admits a big-M free formulation, which can be solved by a branch and cut algorithm. Finally, we present a numerical study to illustrate the effectiveness of the proposed formulations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.07418v4-abstract-full').style.display = 'none'; document.getElementById('1806.07418v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">32 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C15; 90C22; 90C59 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1806.03756">arXiv:1806.03756</a> <span> [<a href="https://arxiv.org/pdf/1806.03756">pdf</a>, <a href="https://arxiv.org/ps/1806.03756">ps</a>, <a href="https://arxiv.org/format/1806.03756">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Scalable Algorithms for the Sparse Ridge Regression </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a>, <a href="/search/math?searchtype=author&query=Deng%2C+X">Xinwei Deng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1806.03756v3-abstract-short" style="display: inline;"> Sparse regression and variable selection for large-scale data have been rapidly developed in the past decades. This work focuses on sparse ridge regression, which enforces the sparsity by use of the L0 norm. We first prove that the continuous relaxation of the mixed integer second order conic (MISOC) reformulation using perspective formulation is equivalent to that of the convex integer formulatio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.03756v3-abstract-full').style.display = 'inline'; document.getElementById('1806.03756v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.03756v3-abstract-full" style="display: none;"> Sparse regression and variable selection for large-scale data have been rapidly developed in the past decades. This work focuses on sparse ridge regression, which enforces the sparsity by use of the L0 norm. We first prove that the continuous relaxation of the mixed integer second order conic (MISOC) reformulation using perspective formulation is equivalent to that of the convex integer formulation proposed in recent work. We also show that the convex hull of the constraint system of MISOC formulation is equal to its continuous relaxation. Based upon these two formulations (i.e., the MISOC formulation and convex integer formulation), we analyze two scalable algorithms, the greedy and randomized algorithms, for sparse ridge regression with desirable theoretical properties. The proposed algorithms are proved to yield near-optimal solutions under mild conditions. We further propose to integrate the greedy algorithm with the randomized algorithm, which can greedily search the features from the nonzero subset identified by the continuous relaxation of the MISOC formulation. The merits of the proposed methods are illustrated through numerical examples in comparison with several existing ones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.03756v3-abstract-full').style.display = 'none'; document.getElementById('1806.03756v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 62J07; 90C10; 90C15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1705.09405">arXiv:1705.09405</a> <span> [<a href="https://arxiv.org/pdf/1705.09405">pdf</a>, <a href="https://arxiv.org/format/1705.09405">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Approximation of Ruin Probabilities via Erlangized Scale Mixtures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Peralta%2C+O">Oscar Peralta</a>, <a href="/search/math?searchtype=author&query=Rojas-Nandayapa%2C+L">Leonardo Rojas-Nandayapa</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wangyue Xie</a>, <a href="/search/math?searchtype=author&query=Yao%2C+H">Hui Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1705.09405v1-abstract-short" style="display: inline;"> In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cram茅r--Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a dense class of distributions that we denominate Erlangized scale mixtures (ESM) and correspond to nonnegative and absolutely continuous distributions which can be writ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.09405v1-abstract-full').style.display = 'inline'; document.getElementById('1705.09405v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1705.09405v1-abstract-full" style="display: none;"> In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cram茅r--Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a dense class of distributions that we denominate Erlangized scale mixtures (ESM) and correspond to nonnegative and absolutely continuous distributions which can be written as a Mellin--Stieltjes convolution $螤\star G$ of a nonnegative distribution $螤$ with an Erlang distribution $G$. A distinctive feature of such a class is that it contains heavy-tailed distributions. We suggest a simple methodology for constructing a sequence of distributions having the form $螤\star G$ to approximate the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.09405v1-abstract-full').style.display = 'none'; document.getElementById('1705.09405v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1502.01811">arXiv:1502.01811</a> <span> [<a href="https://arxiv.org/pdf/1502.01811">pdf</a>, <a href="https://arxiv.org/ps/1502.01811">ps</a>, <a href="https://arxiv.org/format/1502.01811">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Asymptotic tail behavior of phase-type scale mixture distributions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Rojas-Nandayapa%2C+L">Leonardo Rojas-Nandayapa</a>, <a href="/search/math?searchtype=author&query=Xie%2C+W">Wangyue Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1502.01811v2-abstract-short" style="display: inline;"> We consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable $Y$ and a nonnegative but otherwise arbitrary random variable $S$ called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and dete… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.01811v2-abstract-full').style.display = 'inline'; document.getElementById('1502.01811v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1502.01811v2-abstract-full" style="display: none;"> We consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable $Y$ and a nonnegative but otherwise arbitrary random variable $S$ called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction. Particular focus is given to phase-type scale mixture distributions where the scaling random variable $S$ has discrete support --- such a class of distributions has been recently used in risk applications to approximate heavy-tailed distributions. Our results are complemented with several examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.01811v2-abstract-full').style.display = 'none'; document.getElementById('1502.01811v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 0 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1401.2415">arXiv:1401.2415</a> <span> [<a href="https://arxiv.org/pdf/1401.2415">pdf</a>, <a href="https://arxiv.org/format/1401.2415">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Optimal Layout of Transshipment Facilities on An Infinite Homogeneous Plane </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+W">Weijun Xie</a>, <a href="/search/math?searchtype=author&query=Ouyang%2C+Y">Yanfeng Ouyang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1401.2415v2-abstract-short" style="display: inline;"> This paper studies optimal spatial layout of transshipment facilities and the corresponding service regions on an infinite homogeneous plane $\mathbb{R}^2$ that minimize the total cost for facility set-up, outbound delivery and inbound replenishment transportation. The problem has strong implications in the context of freight logistics and transit system design. This paper first focuses on a Eucli… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.2415v2-abstract-full').style.display = 'inline'; document.getElementById('1401.2415v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1401.2415v2-abstract-full" style="display: none;"> This paper studies optimal spatial layout of transshipment facilities and the corresponding service regions on an infinite homogeneous plane $\mathbb{R}^2$ that minimize the total cost for facility set-up, outbound delivery and inbound replenishment transportation. The problem has strong implications in the context of freight logistics and transit system design. This paper first focuses on a Euclidean plane and presents a new proof for the known Gersho's conjecture, which states that the optimal shape of each service region should be a regular hexagon if the inbound transportation cost is ignored. When inbound transportation cost becomes non-negligible, however, we show that a tight upper bound can be achieved by a type of elongated cyclic hexagons, while a cost lower bound based on relaxation and idealization is also obtained. The gap between the analytical upper and lower bounds is within 0.3%. This paper then shows that a similar elongated non-cyclic hexagon shape is actually optimal for service regions on a rectilinear metric plane. Numerical experiments and sensitivity analyses are conducted to verify the analytical findings and to draw managerial insights. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.2415v2-abstract-full').style.display = 'none'; document.getElementById('1401.2415v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 January, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 12 figures</span> </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" 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