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</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/2408.05778">arXiv:2408.05778</a> <span> [<a href="https://arxiv.org/pdf/2408.05778">pdf</a>, <a href="https://arxiv.org/format/2408.05778">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"> Pareto Front Shape-Agnostic Pareto Set Learning in Multi-Objective Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongguang Ye</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Longcan Chen</a>, <a href="/search/math?searchtype=author&query=Kou%2C+W">Wei-Bin Kou</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinyuan Zhang</a>, <a href="/search/math?searchtype=author&query=Ishibuchi%2C+H">Hisao Ishibuchi</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="2408.05778v1-abstract-short" style="display: inline;"> Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high pe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05778v1-abstract-full').style.display = 'inline'; document.getElementById('2408.05778v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.05778v1-abstract-full" style="display: none;"> Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high performance of the PSL methods. Designing a sampling strategy of preference vectors is difficult since the Pareto front shape cannot be known in advance. To make Pareto set learning work effectively in any Pareto front shape, we propose a Pareto front shape-agnostic Pareto Set Learning (GPSL) that does not require the prior information about the Pareto front. The fundamental concept behind GPSL is to treat the learning of the Pareto set as a distribution transformation problem. Specifically, GPSL can transform an arbitrary distribution into the Pareto set distribution. We demonstrate that training a neural network by maximizing hypervolume enables the process of distribution transformation. Our proposed method can handle any shape of the Pareto front and learn the Pareto set without requiring prior knowledge. Experimental results show the high performance of our proposed method on diverse test problems compared with recent Pareto set learning algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05778v1-abstract-full').style.display = 'none'; document.getElementById('2408.05778v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">7 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> IEEE International Conference on Systems, Man, and Cybernetics (IEEE SMC 2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.09736">arXiv:2104.09736</a> <span> [<a href="https://arxiv.org/pdf/2104.09736">pdf</a>, <a href="https://arxiv.org/format/2104.09736">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Neural and Evolutionary Computing">cs.NE</span> <span class="tag is-small is-grey 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/10.1109/TEVC.2021.3093114">10.1109/TEVC.2021.3093114 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Hypervolume-Optimal $渭$-Distributions on Line/Plane-based Pareto Fronts in Three Dimensions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Shang%2C+K">Ke Shang</a>, <a href="/search/math?searchtype=author&query=Ishibuchi%2C+H">Hisao Ishibuchi</a>, <a href="/search/math?searchtype=author&query=Chen%2C+W">Weiyu Chen</a>, <a href="/search/math?searchtype=author&query=Nan%2C+Y">Yang Nan</a>, <a href="/search/math?searchtype=author&query=Liao%2C+W">Weiduo Liao</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="2104.09736v1-abstract-short" style="display: inline;"> Hypervolume is widely used in the evolutionary multi-objective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with $渭$ solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume optimal $渭$-distr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.09736v1-abstract-full').style.display = 'inline'; document.getElementById('2104.09736v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.09736v1-abstract-full" style="display: none;"> Hypervolume is widely used in the evolutionary multi-objective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with $渭$ solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume optimal $渭$-distribution. Theoretical results have shown that the $渭$ solutions are uniformly distributed on a linear Pareto front in two dimensions. However, the $渭$ solutions are not always uniformly distributed on a single-line Pareto front in three dimensions. They are only uniform when the single-line Pareto front has one constant objective. In this paper, we further investigate the hypervolume optimal $渭$-distribution in three dimensions. We consider the line- and plane-based Pareto fronts. For the line-based Pareto fronts, we extend the single-line Pareto front to two-line and three-line Pareto fronts, where each line has one constant objective. For the plane-based Pareto fronts, the linear triangular and inverted triangular Pareto fronts are considered. First, we show that the $渭$ solutions are not always uniformly distributed on the line-based Pareto fronts. The uniformity depends on how the lines are combined. Then, we show that a uniform solution set on the plane-based Pareto front is not always optimal for hypervolume maximization. It is locally optimal with respect to a $(渭+1)$ selection scheme. Our results can help researchers in the community to better understand and utilize the hypervolume indicator. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.09736v1-abstract-full').style.display = 'none'; document.getElementById('2104.09736v1-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> 19 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">This paper has been submitted to a journal for review</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.06773">arXiv:1805.06773</a> <span> [<a href="https://arxiv.org/pdf/1805.06773">pdf</a>, <a href="https://arxiv.org/format/1805.06773">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/10.1109/TEVC.2019.2909271">10.1109/TEVC.2019.2909271 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> R2-based Hypervolume Contribution Approximation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Shang%2C+K">Ke Shang</a>, <a href="/search/math?searchtype=author&query=Ishibuchi%2C+H">Hisao Ishibuchi</a>, <a href="/search/math?searchtype=author&query=Ni%2C+X">Xizi Ni</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="1805.06773v3-abstract-short" style="display: inline;"> In this letter, a new hypervolume contribution approximation method is proposed which is formulated as an R2 indicator. The basic idea of the proposed method is to use different line segments only in the hypervolume contribution region for the hypervolume contribution approximation. Comparing with a traditional method which is based on the R2 indicator to approximate the hypervolume, the new metho… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.06773v3-abstract-full').style.display = 'inline'; document.getElementById('1805.06773v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.06773v3-abstract-full" style="display: none;"> In this letter, a new hypervolume contribution approximation method is proposed which is formulated as an R2 indicator. The basic idea of the proposed method is to use different line segments only in the hypervolume contribution region for the hypervolume contribution approximation. Comparing with a traditional method which is based on the R2 indicator to approximate the hypervolume, the new method can directly approximate the hypervolume contribution and will utilize all the direction vectors only in the hypervolume contribution region. The new method, the traditional method and the Monte Carlo sampling method together with two exact methods are compared through comprehensive experiments. Our results show the advantages of the new method over the other methods. Comparing with the other two approximation methods, the new method achieves the best performance for comparing hypervolume contributions of different solutions and identifying the solution with the smallest hypervolume contribution. Comparing with the exact methods, the new method is computationally efficient in high-dimensional spaces where the exact methods are impractical to use. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.06773v3-abstract-full').style.display = 'none'; document.getElementById('1805.06773v3-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 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </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" 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