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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Meng, K"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <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/2409.02585">arXiv:2409.02585</a> <span> [<a href="https://arxiv.org/pdf/2409.02585">pdf</a>, <a href="https://arxiv.org/ps/2409.02585">ps</a>, <a href="https://arxiv.org/format/2409.02585">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Four fault-free $B_{n-2}$'s in $B_{n}$ under the random node fault model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meng%2C+K">KaiYue Meng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yuxing 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="2409.02585v1-abstract-short" style="display: inline;"> Let $n\geq 4$. Each $B_{n-2}$ in $B_n$ has one of the forms $a_1a_2X^{n-2}$, $a_1X^{n-2}a_2$ and $X^{n-2}a_1a_2$. Let $1-p$ be the fault probiability of each node in the $n$-dimensional bubble-sort network $B_{n}$ under the random node fault model. In this paper, we determine the probability that there are four distinct fault-free $B_{n-2}$'s in $B_{n}$ by considering all possible combinatorial ca… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02585v1-abstract-full').style.display = 'inline'; document.getElementById('2409.02585v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.02585v1-abstract-full" style="display: none;"> Let $n\geq 4$. Each $B_{n-2}$ in $B_n$ has one of the forms $a_1a_2X^{n-2}$, $a_1X^{n-2}a_2$ and $X^{n-2}a_1a_2$. Let $1-p$ be the fault probiability of each node in the $n$-dimensional bubble-sort network $B_{n}$ under the random node fault model. In this paper, we determine the probability that there are four distinct fault-free $B_{n-2}$'s in $B_{n}$ by considering all possible combinatorial cases of the four fault-free $B_{n-2}$'s. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02585v1-abstract-full').style.display = 'none'; document.getElementById('2409.02585v1-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> 4 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">63 pages, no figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C90 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.16053">arXiv:2406.16053</a> <span> [<a href="https://arxiv.org/pdf/2406.16053">pdf</a>, <a href="https://arxiv.org/format/2406.16053">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"> Lipschitz continuity of solution multifunctions of extended $\ell_1$ regularization problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Wu%2C+P">Pengcheng Wu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="2406.16053v1-abstract-short" style="display: inline;"> In this paper we obtain a verifiable sufficient condition for a polyhedral multifunction to be Lipschitz continuous on its domain. We apply this sufficient condition to establish the Lipschitz continuity of the solution multifunction for an extended $\ell_1$ regularization problem with respect to the regularization parameter and the observation parameter under the assumption that the data matrix i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.16053v1-abstract-full').style.display = 'inline'; document.getElementById('2406.16053v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.16053v1-abstract-full" style="display: none;"> In this paper we obtain a verifiable sufficient condition for a polyhedral multifunction to be Lipschitz continuous on its domain. We apply this sufficient condition to establish the Lipschitz continuity of the solution multifunction for an extended $\ell_1$ regularization problem with respect to the regularization parameter and the observation parameter under the assumption that the data matrix is of full row rank. In doing so, we show that the solution multifunction is a polyhedral one by expressing its graph as the union of the polyhedral cones constructed by the index sets defined by nonempty faces of the feasible set of an extended dual $\ell_1$ regularization problem. We demonstrate that the domain of the solution multifunction is partitioned as the union of the projections of the above polyhedral cones onto the parameters space and that the graph of the restriction of the multifunction on each of these projections is convex. In comparing with the existing result of the local Lipschitz continuity of the Lasso problem in the literature where certain linear independence condition was assumed, our condition (i.e., full row rank of data matrix) is very weak and our result (i.e., Lipschitz continuity on the domain) is much more stronger. As corollaries of the Lipschitz continuity of the solution multifunction, we show that the single-valuedness and linearity (or piecewise linearity) of the solution multifunction on a particular polyhedral set of the domain are equivalent to certain linear independence conditions of the corresponding columns of the data matrix proposed in the literature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.16053v1-abstract-full').style.display = 'none'; document.getElementById('2406.16053v1-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> 23 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">25 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65K05; 90C25; 90C31 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.03142">arXiv:2309.03142</a> <span> [<a href="https://arxiv.org/pdf/2309.03142">pdf</a>, <a href="https://arxiv.org/ps/2309.03142">ps</a>, <a href="https://arxiv.org/format/2309.03142">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Euler Characteristics and Homotopy Types of Definable Sublevel Sets, with Applications to Topological Data Analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ji%2C+M">Mattie Ji</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Kun Meng</a>, <a href="/search/math?searchtype=author&query=Ding%2C+K">Kexin Ding</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="2309.03142v2-abstract-short" style="display: inline;"> Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t = \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+未}$ deformation retracts to $S^f_t$ for all sufficientl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.03142v2-abstract-full').style.display = 'inline'; document.getElementById('2309.03142v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.03142v2-abstract-full" style="display: none;"> Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t = \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+未}$ deformation retracts to $S^f_t$ for all sufficiently small $未> 0$. Applying these results, we also characterize the connections between the following concepts in topological data analysis: the Euler characteristic transform (ECT), smooth ECT, Euler-Radon transform (ERT), and smooth ERT. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.03142v2-abstract-full').style.display = 'none'; document.getElementById('2309.03142v2-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> 4 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">20 page</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 03C64; 46M20. Secondary: 55N31 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.15760">arXiv:2308.15760</a> <span> [<a href="https://arxiv.org/pdf/2308.15760">pdf</a>, <a href="https://arxiv.org/ps/2308.15760">ps</a>, <a href="https://arxiv.org/format/2308.15760">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"> Variational Analysis of Kurdyka-艁ojasiewicz Property, Exponent and Modulus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+M">Minghua Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="2308.15760v2-abstract-short" style="display: inline;"> The Kurdyka-艁ojasiewicz (K艁) property, exponent and modulus have played a very important role in the study of global convergence and rate of convergence for optimal algorithms. In this paper, at a stationary point of a locally lower semicontinuous function, we obtain complete characterizations of the K艁 property and the K艁 modulus via the outer limiting subdifferential of an auxilliary function an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.15760v2-abstract-full').style.display = 'inline'; document.getElementById('2308.15760v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.15760v2-abstract-full" style="display: none;"> The Kurdyka-艁ojasiewicz (K艁) property, exponent and modulus have played a very important role in the study of global convergence and rate of convergence for optimal algorithms. In this paper, at a stationary point of a locally lower semicontinuous function, we obtain complete characterizations of the K艁 property and the K艁 modulus via the outer limiting subdifferential of an auxilliary function and a newly-introduced subderivative function respectively. In particular, for a class of prox-regular, twice epi-differentiable and subdifferentially continuous functions, we show that the K艁 property and the K艁 modulus can be described by its Moreau envelopes and a quadratic growth condition. We apply the obtained results to establish the K艁 property with exponent $\frac12$ and to provide calculation of the modulus for a smooth function, the pointwise maximum of finitely many smooth functions and regularized functions respectively. These functions often appear in the modelling of structured optimization problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.15760v2-abstract-full').style.display = 'none'; document.getElementById('2308.15760v2-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 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">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 49J53; 90C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.02759">arXiv:2212.02759</a> <span> [<a href="https://arxiv.org/pdf/2212.02759">pdf</a>, <a href="https://arxiv.org/ps/2212.02759">ps</a>, <a href="https://arxiv.org/format/2212.02759">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"> Kurdyka-艁ojasiewicz Inequality and Error Bounds of D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+M+H">M. H. Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K+W">K. W. Meng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X+Q">X. Q. 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="2212.02759v1-abstract-short" style="display: inline;"> In this paper, we study the D-gap function associated with a nonsmooth and nonmonotone variational inequality problem. We present some exact formulas for the subderivative, the regular subdifferential set, and the limiting subdifferential set of the D-gap function. By virtue of these formulas, we provide some sufficient and necessary conditions for the Kurdyka-艁ojasiewicz inequality property and t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.02759v1-abstract-full').style.display = 'inline'; document.getElementById('2212.02759v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.02759v1-abstract-full" style="display: none;"> In this paper, we study the D-gap function associated with a nonsmooth and nonmonotone variational inequality problem. We present some exact formulas for the subderivative, the regular subdifferential set, and the limiting subdifferential set of the D-gap function. By virtue of these formulas, we provide some sufficient and necessary conditions for the Kurdyka-艁ojasiewicz inequality property and the error bound property for the D-gap functions. As an application of our Kurdyka-艁ojasiewicz inequality result and the abstract convergence result in [Attouch, et al., Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program., 137(2013)91-129], we show that the sequence generated by a derivative free descent algorithm with an inexact line search converges linearly to some solution of the variational inequality problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.02759v1-abstract-full').style.display = 'none'; document.getElementById('2212.02759v1-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 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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">38 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65K10; 65K15; 90C26; 49M37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.11706">arXiv:2210.11706</a> <span> [<a href="https://arxiv.org/pdf/2210.11706">pdf</a>, <a href="https://arxiv.org/format/2210.11706">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 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/10.1007/s11228-023-00698-9">10.1007/s11228-023-00698-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Projectional Coderivatives and Calculus Rules </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yao%2C+W">Wenfang Yao</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Li%2C+M">Minghua Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="2210.11706v2-abstract-short" style="display: inline;"> This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to give a com… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.11706v2-abstract-full').style.display = 'inline'; document.getElementById('2210.11706v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.11706v2-abstract-full" style="display: none;"> This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to give a complete characterization of the relative Lipschitz-like property under such a setting. Chain rules and sum rules are obtained to facilitate the application of the tool to a wider range of problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.11706v2-abstract-full').style.display = 'none'; document.getElementById('2210.11706v2-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 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 49J53; 49K40; 58C07; 90C31 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.08334">arXiv:2102.08334</a> <span> [<a href="https://arxiv.org/pdf/2102.08334">pdf</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="Materials Science">cond-mat.mtrl-sci</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</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.jsv.2021.116354">10.1016/j.jsv.2021.116354 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A homogenized damping model for the propagation of elastic wave in a porous solid </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meng%2C+K">Kangpei Meng</a>, <a href="/search/math?searchtype=author&query=Li%2C+Q">Qingming Li</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="2102.08334v1-abstract-short" style="display: inline;"> This paper develops an averaging technique based on the combination of the eigenfunction expansion method and the collaboration method to investigate the multiple scattering effect of the SH wave propagation in a porous medium. The semi-analytical averaging technique is conducted using Monto Carlo method to understand the macroscopic dispersion and attenuation phenomena of the stress wave propagat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.08334v1-abstract-full').style.display = 'inline'; document.getElementById('2102.08334v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.08334v1-abstract-full" style="display: none;"> This paper develops an averaging technique based on the combination of the eigenfunction expansion method and the collaboration method to investigate the multiple scattering effect of the SH wave propagation in a porous medium. The semi-analytical averaging technique is conducted using Monto Carlo method to understand the macroscopic dispersion and attenuation phenomena of the stress wave propagation in a porous solid caused by the multiple scattering effects. The averaging technique is verified by finite element analysis. Finally, a simple homogenized elastic model with damping is proposed to describe the macroscopic dispersion and attenuation effects of SH waves in porous media. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.08334v1-abstract-full').style.display = 'none'; document.getElementById('2102.08334v1-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 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">33 pages, 16 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14J60 (Primary) 14F05; 14J26 (Secondary) <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.2.2; I.2.7 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2009.10536">arXiv:2009.10536</a> <span> [<a href="https://arxiv.org/pdf/2009.10536">pdf</a>, <a href="https://arxiv.org/ps/2009.10536">ps</a>, <a href="https://arxiv.org/format/2009.10536">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"> Lipschitz-like property relative to a set and the generalized Mordukhovich criterion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Li%2C+M">Minghua Li</a>, <a href="/search/math?searchtype=author&query=Yao%2C+W">Wenfang Yao</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="2009.10536v1-abstract-short" style="display: inline;"> In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.10536v1-abstract-full').style.display = 'inline'; document.getElementById('2009.10536v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2009.10536v1-abstract-full" style="display: none;"> In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.10536v1-abstract-full').style.display = 'none'; document.getElementById('2009.10536v1-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 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.01573">arXiv:1911.01573</a> <span> [<a href="https://arxiv.org/pdf/1911.01573">pdf</a>, <a href="https://arxiv.org/format/1911.01573">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="Signal Processing">eess.SP</span> </div> </div> <p class="title is-5 mathjax"> Improving Operational Feasibility of Low-voltage Distribution Network by Phase-switching Devices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+B">Bin Liu</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Ke Meng</a>, <a href="/search/math?searchtype=author&query=Wong%2C+P+K+C">Peter K. C. Wong</a>, <a href="/search/math?searchtype=author&query=Dong%2C+Z+Y">Zhao Yang Dong</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+C">Cuo Zhang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+B">Bo Wang</a>, <a href="/search/math?searchtype=author&query=Ting%2C+T">Tian Ting</a>, <a href="/search/math?searchtype=author&query=Qi%2C+Q">Qu Qi</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="1911.01573v1-abstract-short" style="display: inline;"> High penetration of residential photovoltaic (PV) in low-voltage distribution networks (LVDN) makes the unbalance issue more sever due to the asymmetry of generation/load characteristic in different phases, and may lead to infeasible operation of the whole network. Phase switching device (PSD), which can switch the connected phase of a residential load as required, is viable and efficient equipmen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.01573v1-abstract-full').style.display = 'inline'; document.getElementById('1911.01573v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.01573v1-abstract-full" style="display: none;"> High penetration of residential photovoltaic (PV) in low-voltage distribution networks (LVDN) makes the unbalance issue more sever due to the asymmetry of generation/load characteristic in different phases, and may lead to infeasible operation of the whole network. Phase switching device (PSD), which can switch the connected phase of a residential load as required, is viable and efficient equipment to help address this issue. This paper, based on three-phase power flow (TUPF) formulation, aims to investigate the benefit of PSD on improving the operation feasibility of LVDN. The linear model of TUPF with PSD is presented to effectively take the flexible device into account, which can be conveniently used to seek the PSD positions that lead to a viable operation strategy of the original problem when infeasibility is reported by the traditional iteration-based algorithm. Case study based on a practical LVDN in Australia demonstrates the efficacy of the proposed method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.01573v1-abstract-full').style.display = 'none'; document.getElementById('1911.01573v1-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> 4 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">Accepted and presented at The 8th International Conference on Renewable Power Generation (IET RPG 2019)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.11863">arXiv:1909.11863</a> <span> [<a href="https://arxiv.org/pdf/1909.11863">pdf</a>, <a href="https://arxiv.org/format/1909.11863">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"> Unbalance Mitigation via Phase-switching Device and Static Var Compensator in Low-voltage Distribution Network </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+B">Bin Liu</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Ke Meng</a>, <a href="/search/math?searchtype=author&query=Dong%2C+Z+Y">Zhao Yang Dong</a>, <a href="/search/math?searchtype=author&query=Wong%2C+P+K+C">Peter K. C. Wong</a>, <a href="/search/math?searchtype=author&query=Ting%2C+T">Tian Ting</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="1909.11863v4-abstract-short" style="display: inline;"> As rooftop solar PVs installed by residential customers penetrate in low voltage distribution network (LVDN), some issues, e.g. over/under voltage and unbalances, which may undermine the network's operational performance, need to be effectively addressed. To mitigate unbalances in LVDN, dynamic switching devices (PSDs) and static var compensator (SVC) are two equipment that are cost-effective and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.11863v4-abstract-full').style.display = 'inline'; document.getElementById('1909.11863v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.11863v4-abstract-full" style="display: none;"> As rooftop solar PVs installed by residential customers penetrate in low voltage distribution network (LVDN), some issues, e.g. over/under voltage and unbalances, which may undermine the network's operational performance, need to be effectively addressed. To mitigate unbalances in LVDN, dynamic switching devices (PSDs) and static var compensator (SVC) are two equipment that are cost-effective and efficient. However, most existing research on operating PSDs are based on inflexible heuristic algorithms or without considering the network formulation, which may lead to strategies that violate operational requirements. Moreover, few pieces of literature have been reported on mitigating unbalances in LVDN via SVC and PSDs together. This paper, after presenting the dispatch model of SVC, formulates the decision-making process as a mixed-integer non-convex programming (MINCP) problem considering all practical operational requirements. To efficiently solve the challenging problem, the MINCP is reformulated as a mixed-integer second order-cone programming (MISOCP) problem based on either exact reformulations or accurate approximations, making it possible to employ efficient off-the-shelf solvers. Simulations based on a modified IEEE system and a practical system in Australia demonstrates the efficiency of the proposed method in mitigating unbalances in LVDN. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.11863v4-abstract-full').style.display = 'none'; document.getElementById('1909.11863v4-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> 29 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">8 Pages, 12 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/1708.07010">arXiv:1708.07010</a> <span> [<a href="https://arxiv.org/pdf/1708.07010">pdf</a>, <a href="https://arxiv.org/ps/1708.07010">ps</a>, <a href="https://arxiv.org/format/1708.07010">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"> Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yaohua Hu</a>, <a href="/search/math?searchtype=author&query=Li%2C+C">Chong Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="1708.07010v1-abstract-short" style="display: inline;"> The $\ell_p$ regularization problem with $0< p< 1$ has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. The proximal gradient algorithm is one of the most popular algorithms for solving the $\ell_p$ regularisation problem. In the present paper, we investigate the linear convergence issu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.07010v1-abstract-full').style.display = 'inline'; document.getElementById('1708.07010v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1708.07010v1-abstract-full" style="display: none;"> The $\ell_p$ regularization problem with $0< p< 1$ has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. The proximal gradient algorithm is one of the most popular algorithms for solving the $\ell_p$ regularisation problem. In the present paper, we investigate the linear convergence issue of one inexact descent method and two inexact proximal gradient algorithms (PGA). For this purpose, an optimality condition theorem is explored to provide the equivalences among a local minimum, second-order optimality condition and second-order growth property of the $\ell_p$ regularization problem. By virtue of the second-order optimality condition and second-order growth property, we establish the linear convergence properties of the inexact descent method and inexact PGAs under some simple assumptions. Both linear convergence to a local minimal value and linear convergence to a local minimum are provided. Finally, the linear convergence results of the inexact numerical methods are extended to the infinite-dimensional Hilbert spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.07010v1-abstract-full').style.display = 'none'; document.getElementById('1708.07010v1-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> 23 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2017. </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">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65K05; 65J22 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1608.03360">arXiv:1608.03360</a> <span> [<a href="https://arxiv.org/pdf/1608.03360">pdf</a>, <a href="https://arxiv.org/ps/1608.03360">ps</a>, <a href="https://arxiv.org/format/1608.03360">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 Error Bound Moduli for Locally Lipschitz and Regular Functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+M">Minghua Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="1608.03360v1-abstract-short" style="display: inline;"> In this paper we study local error bound moduli for a locally Lipschitz and regular function via its outer limiting subdifferential set. We show that the distance of 0 from the outer limiting subdifferential of the support function of the subdifferential set, which is essentially the distance of 0 from the end set of the subdifferential set, is an upper estimate of the local error bound modulus. T… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.03360v1-abstract-full').style.display = 'inline'; document.getElementById('1608.03360v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1608.03360v1-abstract-full" style="display: none;"> In this paper we study local error bound moduli for a locally Lipschitz and regular function via its outer limiting subdifferential set. We show that the distance of 0 from the outer limiting subdifferential of the support function of the subdifferential set, which is essentially the distance of 0 from the end set of the subdifferential set, is an upper estimate of the local error bound modulus. This upper estimate becomes tight for a convex function under some regularity conditions. We show that the distance of 0 from the outer limiting subdifferential set of a lower $\mathcal{C}^1$ function is equal to the local error bound modulus. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.03360v1-abstract-full').style.display = 'none'; document.getElementById('1608.03360v1-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> 10 August, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2016. </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">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1606.05710">arXiv:1606.05710</a> <span>  </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 Novel Projected Two Binary Variables Formulation for Unit Commitment Problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+L">Linfeng Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+C">Chen Zhang</a>, <a href="/search/math?searchtype=author&query=Jian%2C+J">Jinbao Jian</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Ke Meng</a>, <a href="/search/math?searchtype=author&query=Dong%2C+Z">Zhaoyang Dong</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="1606.05710v2-abstract-short" style="display: inline;"> The thermal unit commitment (UC) problem often can be formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale instances. In this paper, with projecting unit generation level onto [0,1] and reformulation techniques, a novel two binary (2-bin) variables MIQP formulation for UC problem is presented. We show that 2-bin formulation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1606.05710v2-abstract-full').style.display = 'inline'; document.getElementById('1606.05710v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1606.05710v2-abstract-full" style="display: none;"> The thermal unit commitment (UC) problem often can be formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale instances. In this paper, with projecting unit generation level onto [0,1] and reformulation techniques, a novel two binary (2-bin) variables MIQP formulation for UC problem is presented. We show that 2-bin formulation is more compact than the state-of-the-art one binary (1-bin) variable formulation and three binary (3-bin) variables formulation. Moreover, 2-bin formulation is tighter than 1-bin and 3-bin formulations in quadratic cost function, and it is tighter than 1-bin formulation in linear constraints. Three mixed integer linear programming (MILP) formulations can be obtained from three UC MIQPs by replacing the quadratic terms in the objective functions by a sequence of piece-wise perspective-cuts. 2-bin MILP is also the best one due to the similar reasons of MIQP. The simulation results for realistic instances that range in size from 10 to 200 units over a scheduling period of 24 hours show that the proposed 2-bin formulations are competitive with currently state-of-the-art formulations and promising for large-scale UC problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1606.05710v2-abstract-full').style.display = 'none'; document.getElementById('1606.05710v2-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 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 June, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2016. </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">1.Because of lack of language expression, so we want to improve both language and organization quality. 2.in order to the projected two-binary-variable formulation could be used for the real applications. we should added further analyses and numerical tests.3.In line 75, section 3, The lack of proof</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.07779">arXiv:1601.07779</a> <span> [<a href="https://arxiv.org/pdf/1601.07779">pdf</a>, <a href="https://arxiv.org/ps/1601.07779">ps</a>, <a href="https://arxiv.org/format/1601.07779">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"> Group sparse optimization via $\ell_{p,q}$ regularization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yaohua Hu</a>, <a href="/search/math?searchtype=author&query=Li%2C+C">Chong Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+K">Kaiwen Meng</a>, <a href="/search/math?searchtype=author&query=Qin%2C+J">Jing Qin</a>, <a href="/search/math?searchtype=author&query=Yang%2C+X">Xiaoqi 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="1601.07779v1-abstract-short" style="display: inline;"> In this paper, we investigate a group sparse optimization problem via $\ell_{p,q}$ regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish some oracle property and a global recovery bound of order $O(位^\frac{2}{2-q})$ for any point in a level set of the $\ell_{p,q}$ regularization… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.07779v1-abstract-full').style.display = 'inline'; document.getElementById('1601.07779v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.07779v1-abstract-full" style="display: none;"> In this paper, we investigate a group sparse optimization problem via $\ell_{p,q}$ regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish some oracle property and a global recovery bound of order $O(位^\frac{2}{2-q})$ for any point in a level set of the $\ell_{p,q}$ regularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order $O(位^2)$ for a path of local minima. In the algorithmic aspect, we apply the well-known proximal gradient method to solve the $\ell_{p,q}$ regularization problems, either by analytically solving some specific $\ell_{p,q}$ regularization subproblems, or by using the Newton method to solve general $\ell_{p,q}$ regularization subproblems. In particular, we establish the linear convergence rate of the proximal gradient method for solving the $\ell_{1,q}$ regularization problem under some mild conditions. As a consequence, the linear convergence rate of proximal gradient method for solving the usual $\ell_{q}$ regularization problem ($0<q<1$) is obtained. Finally in the aspect of application, we present some numerical results on both the simulated data and the real data in gene transcriptional regulation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.07779v1-abstract-full').style.display = 'none'; document.getElementById('1601.07779v1-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 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2016. </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">48 pages, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65K05; 49M37; 90C26 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> B.1.5 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 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