Document Zbl 07923872

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Methods Oper. Res.</a> <a href="/?q=in%3A518812" title="Articles in this Issue">100, No. 1, 263-290 (2024)</a>. </div> <div class="abstract">Summary: In this paper, we present an outer approximation algorithm for computing the Edgeworth-Pareto hull of multi-objective mixed-integer linear programming problems (MOMILPs). It produces the extreme points (i.e., the vertices) as well as the facets of the Edgeworth-Pareto hull. We note that these extreme points are the extreme supported non-dominated points of a MOMILP. We also show how to extend the concept of geometric duality for multi-objective linear programming problems to the Edgeworth-Pareto hull of MOMILPs and use this extension to develop the algorithm. The algorithm relies on a novel oracle that solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the extreme supported non-dominated points in the case of MOMILPs. Thus, for MOMILPs for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay – a result that was formerly only known for the computation of extreme supported non-dominated points. Our algorithm can be an attractive option to compute lower bound sets within multi-objective branch-and-bound algorithms for solving MOMILPs. This is for several reasons as (i) the algorithm starts from a trivial valid lower bound set then iteratively improves it, thus at any iteration of the algorithm a lower bound set is available; (ii) the algorithm also produces efficient solutions (i.e., solutions in the decision space); (iii) in any iteration of the algorithm, a relaxation of the MOMILP can be solved, and the obtained points and facets still provide a valid lower bound set. Moreover, for the special case of multi-objective linear programming problems, the algorithm solves the problem to global optimality. A computational study on a set of benchmark instances from the literature is provided.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7923872">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C29" title="MSC2020">90C29</a> </td> <td class="space"> Multi-objective and goal programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C11" title="MSC2020">90C11</a> </td> <td class="space"> Mixed integer programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C59" title="MSC2020">90C59</a> </td> <td class="space"> Approximation methods and heuristics in mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amulti-objective+optimization">multi-objective optimization</a>; <a href="/?q=ut%3Aouter+approximation">outer approximation</a>; <a href="/?q=ut%3Amixed+integer+programming">mixed integer programming</a></div> <div class="software"> <h3>Software:</h3><a href="/software/15158">PolySCIP</a>; <a href="/software/12273">BENSOLVE</a>; <a href="/software/5357">PPL</a>; <a href="/software/1091">SCIP</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 07923872" data-ciurl="/ci/07923872" data-biburl="/bibtex/07923872.bib" data-amsurl="/amsrefs/07923872.bib" data-xmlurl="/xml/07923872.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07923872.pdf" title="Zbl 07923872 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00186-023-00847-8" aria-label="DOI for “An outer approximation algorithm for generating the Edgeworth-Pareto hull of multi-objective mixed-integer linear programming problems”" title="10.1007/s00186-023-00847-8">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2103.16647"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adelgren, N.; Gupte, A., Branch-and-bound for biobjective mixed-integer linear programming, INFORMS J Comput, 34, 2, 909-933, 2022 · <a href="/07551217" class="nowrap">Zbl 07551217</a></td> </tr><tr> <td>[2]</td> <td class="space">Aneja, YP; Nair, KP, Bicriteria transportation problem, Manag Sci, 25, 1, 73-78, 1979 · <a href="/0442.90056" class="nowrap">Zbl 0442.90056</a></td> </tr><tr> <td>[3]</td> <td class="space">Applegate D, Bixby R, Chvátal V, Cook W (2001) TSP cuts which do not conform to the template paradigm. 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