Document Zbl 1257.90077

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Glob. Optim.</a> <a href="/?q=in%3A293603" title="Articles in this Issue">52, No. 1, 1-28 (2012)</a>. </div> <div class="abstract"><div class="pre">Summary: Theory for the convergence order of the convex relaxations by <span class="zbmathjax-textit">G. P. McCormick</span> [Math. Program. 10, 147–175 (1976; <a href="/0349.90100">Zbl 0349.90100</a>)] for factorable functions is developed. Convergence rules are established for the addition, multiplication and composition operations. The convergence order is considered both in terms of pointwise convergence and of convergence in the Hausdorff metric. The convergence order of the composite function depends on the convergence order of the relaxations of the factors. No improvement in the order of convergence compared to that of the underlying bound calculation, e.g., via interval extensions, can be guaranteed unless the relaxations of the factors have pointwise convergence of high order. The McCormick relaxations are compared with the $\alpha $BB relaxations by Floudas and coworkers [<span class="zbmathjax-textit">C. D. Maranas</span>, “A global optimization approach for Lennard-Jones microclusters”, J. Chem. Phys. 97, No. 10, 7667–7678 (1992; <a href="https://doi.org/10.1063/1.463486">doi:10.1063/1.463486</a>); <span class="zbmathjax-textit">C. D. Maranas</span>, J. Glob. Optim. 7, No. 2, 143–182 (199; <a href="/0841.90115">Zbl 0841.90115</a>); <span class="zbmathjax-textit">C. S. Adjiman</span>, J. Glob. Optim. 9, No. 1, 23–40 (1996; <a href="/0862.90114">Zbl 0862.90114</a>)], which guarantee quadratic convergence. Illustrative and numerical examples are given.</div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5997547">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5997547">Cited in <strong>36</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C26" title="MSC2020">90C26</a> </td> <td class="space"> Nonconvex programming, global optimization </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aconvex+relaxation">convex relaxation</a>; <a href="/?q=ut%3AMcCormick">McCormick</a>; <a href="/?q=ut%3AalphaBB">alphaBB</a>; <a href="/?q=ut%3Ainterval+extensions">interval extensions</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0349.90100">Zbl 0349.90100</a>; <a href="/0841.90115">Zbl 0841.90115</a>; <a href="/0862.90114">Zbl 0862.90114</a></div> <div class="software"> <h3>Software:</h3><a href="/software/6249">alphaBB</a>; <a href="/software/66">BARON</a>; <a href="/software/6615">libMC</a>; <a href="/software/13199">MC++</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 1257.90077" data-ciurl="/ci/05997547" data-biburl="/bibtex/05997547.bib" data-amsurl="/amsrefs/05997547.bib" data-xmlurl="/xml/05997547.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05997547.pdf" title="Zbl 1257.90077 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/s10898-011-9685-2" aria-label="DOI for “Convergence rate of McCormick relaxations”" title="10.1007/s10898-011-9685-2">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adjiman C.S., Androulakis I.P., Floudas C.A.: A global optimization method, {$\alpha$}BB for general twice-differentiable constrained NLPs–II. 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