Document Zbl 08018416

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Optim. Appl.</a> <a href="/?q=in%3A526946" title="Articles in this Issue">90, No. 3, 911-939 (2025)</a>. </div> <div class="abstract">Summary: In the last two decades, the sequential optimality conditions, which do not require constraint qualifications and allow improvement on the convergence assumptions of algorithms, had been considered in the literature. It includes the work by Andreani et al. (IMA J Numer Anal 37:1902-1929, 2017), with a sequential optimality condition for nonlinear programming, that uses the second-order information of the problem. More recently, Fukuda et al. (Set-Valued Var Anal 31:15, 2023) analyzed the conditions that use second-order information, in particular for nonlinear second-order cone programming problems (SOCP). However, such optimality conditions were not defined explicitly. In this paper, we propose an explicit definition of approximate-Karush-Kuhn-Tucker 2 (AKKT2) and complementary-AKKT2 (CAKKT2) conditions for SOCPs. We prove that the proposed AKKT2/CAKKT2 conditions are satisfied at local optimal points of the SOCP without any constraint qualification. We also present two algorithms that are based on augmented Lagrangian and sequential quadratic programming methods and show their global convergence to points satisfying the proposed conditions.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90Cxx" title="MSC2020">90Cxx</a> </td> <td class="space"> Mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aoptimality+conditions">optimality conditions</a>; <a href="/?q=ut%3Asecond-order+optimality">second-order optimality</a>; <a href="/?q=ut%3Asecond-order+cone+programming">second-order cone programming</a>; <a href="/?q=ut%3Aconic+optimization">conic optimization</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 08018416" data-ciurl="/ci/08018416" data-biburl="/bibtex/08018416.bib" data-amsurl="/amsrefs/08018416.bib" data-xmlurl="/xml/08018416.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/08018416.pdf" title="Zbl 08018416 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/s10589-025-00649-0" aria-label="DOI for “A second-order sequential optimality condition for nonlinear second-order cone programming problems”" title="10.1007/s10589-025-00649-0">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2310.01971"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">Lobo, MS; Vandenberghe, L.; Boyd, S.; Lebret, H., Applications of second-order cone programming, Linear Algebra Appl., 284, 1, 193-228, 1998 · <a href="/0946.90050" class="nowrap">Zbl 0946.90050</a> · <a href="https://doi.org/10.1016/S0024-3795(98)10032-0" class="nowrap">doi:10.1016/S0024-3795(98)10032-0</a></td> </tr><tr> <td>[2]</td> <td class="space">Fukuda, EH; Fukushima, M., The use of squared slack variables in nonlinear second-order cone programming, J. 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