Document Zbl 1449.90325

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Ind. Manag. Optim.</a> <a href="/?q=in%3A450955" title="Articles in this Issue">16, No. 2, 1009-1035 (2020)</a>. </div> <div class="abstract">Summary: With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point if the original problem is feasible, and it can be superlinearly or quadratically convergent to the infeasible stationary point when the problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems, where the superlinear convergence to an infeasible stationary point is demonstrated when we solve two infeasible problems in the literature.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7221688">Cited in <strong>11</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C30" title="MSC2020">90C30</a> </td> <td class="space"> Nonlinear programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C51" title="MSC2020">90C51</a> </td> <td class="space"> Interior-point methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C26" title="MSC2020">90C26</a> </td> <td class="space"> Nonconvex programming, global optimization </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C46" title="MSC2020">90C46</a> </td> <td class="space"> Optimality conditions and duality in mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlinear+programming">nonlinear programming</a>; <a href="/?q=ut%3Aconstrained+optimization">constrained optimization</a>; <a href="/?q=ut%3Ainfeasibility">infeasibility</a>; <a href="/?q=ut%3Ainterior-point+method">interior-point method</a>; <a href="/?q=ut%3Aglobal+and+local+convergence">global and local convergence</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4010">CUTEr</a>; <a href="/software/4778">ipfilter</a>; <a href="/software/4808">Ipopt</a>; <a href="/software/4712">LIPSOL</a>; <a href="/software/14681">CUTE</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 1449.90325" data-ciurl="/ci/07221688" data-biburl="/bibtex/07221688.bib" data-amsurl="/amsrefs/07221688.bib" data-xmlurl="/xml/07221688.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07221688.pdf" title="Zbl 1449.90325 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/10.3934/jimo.2018190" aria-label="DOI for “A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs”" title="10.3934/jimo.2018190">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1807.02900"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">R. 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