Document Zbl 0742.90056

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II.</strong> <i>(English)</i> <a class="label nowrap" href="/0742.90056">Zbl 0742.90056</a> </h2> <div class="source"> <a href="/serials/1746" title="Journal Profile">Math. Program., Ser. B</a> <a href="/?q=in%3A158534" title="Articles in this Issue">52, No. 3, 527-553 (1991)</a>. </div> <div class="abstract">Summary: [For part I see the authors, in: Operations Research, Proc. 14th Symp., Ulm/FRG 1989, Methods Oper. Res. 62, 19-31 (1990; <a href="/0726.90056">Zbl 0726.90056</a>).]<br class="zbmathjax-paragraph">A class of algorithms is proposed for solving linear programming problems (with $m$ inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newton’s method to compute points on the central path $x(r)$, $r>0$, and this allows to estimate the complexity, i.e. the total number $N=N(R,\delta)$ of steps needed to go from an initial point $x(R)$ to a final point $x(\delta)$, $R>\delta>0$, by an integral of the local “weighted curvature” of the (primal-dual) path. Here, the central curve is parametrized with the logarithmic penalty parameter $r\downarrow 0$. It is shown that for large classes of problems the complexity integral, i.e. the number of steps $N$, is not greater than const $m^ \alpha\log(R/\delta)$, where $\alpha<{1\over 2}$ e.g. $\alpha={1\over 4}$ or $\alpha={3\over 8}$ (note that $\alpha={1\over 2}$ gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for $\alpha\geq{1\over 3}$.<br class="zbmathjax-paragraph">As a byproduct, many analytical and structural properties of the primal- dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameter $r$ is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A29029">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A29029">Cited in <strong>37</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C05" title="MSC2020">90C05</a> </td> <td class="space"> Linear programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90-08" title="MSC2020">90-08</a> </td> <td class="space"> Computational methods for problems pertaining to operations research and mathematical programming </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C60" title="MSC2020">90C60</a> </td> <td class="space"> Abstract computational complexity for mathematical programming problems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ainequality+constraints">inequality constraints</a>; <a href="/?q=ut%3ANewton%27s+method">Newton’s method</a>; <a href="/?q=ut%3Acentral+curve">central curve</a>; <a href="/?q=ut%3Aweighted+curvature">weighted curvature</a>; <a href="/?q=ut%3Alogarithmic+derivatives+of+the+slack+variables">logarithmic derivatives of the slack variables</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0726.90056">Zbl 0726.90056</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 0742.90056" data-ciurl="/ci/00029029" data-biburl="/bibtex/00029029.bib" data-amsurl="/amsrefs/00029029.bib" data-xmlurl="/xml/00029029.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00029029.pdf" title="Zbl 0742.90056 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/BF01582904" aria-label="DOI for “On the complexity of following the central path of linear programs by linear extrapolation. II”" title="10.1007/BF01582904">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">I. Adler, N. Karmarkar, M.G.C. Resende and G. Veiga, ”An implementation of Karmarkar’s algorithm for linear programming,”Mathematical Programming 44 (1989) 297–335. · <a href="/0682.90061" class="nowrap">Zbl 0682.90061</a> · <a href="https://doi.org/10.1007/BF01587095" class="nowrap">doi:10.1007/BF01587095</a></td> </tr><tr> <td>[2]</td> <td class="space">W. Blaschke and K. Reidmeister,Affine Differentialgeometrie (Springer, Berlin, 1924).</td> </tr><tr> <td>[3]</td> <td class="space">P.T. Boggs, P.D. Domich, J.R. Donaldson and C. Witzgall, ”Algorithmic enhancements to the method of centers for linear programming,”ORSA Journal on Computing 1 (1989) 159–171. · <a href="/0753.90040" class="nowrap">Zbl 0753.90040</a></td> </tr><tr> <td>[4]</td> <td class="space">P.T. Boggs, P.D. Domich, J.R. Donaldson and C. Witzgall, ”Optimal 3-dimensional methods for linear programming,” Tehcnical Report NISTIR 89-4225, National Institute of Standards and Technology (Gaithersburg, MD, 1989). · <a href="/0753.90040" class="nowrap">Zbl 0753.90040</a></td> </tr><tr> <td>[5]</td> <td class="space">J.L. Goffin, A. Haurie and J.P. Vial, ”Decomposition and nondifferentiable optimization with the projective algorithm,” Preprint G-89-25, Faculty of Manamgement, McGill University (Montreal, Que., 1989). · <a href="/0762.90050" class="nowrap">Zbl 0762.90050</a></td> </tr><tr> <td>[6]</td> <td class="space">Cl. Gonzaga, ”An algorithm for solving linear programming problems in O(n 3 L) operations,” in: N. Meggido, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer, New York, 1988) pp. 1–28.</td> </tr><tr> <td>[7]</td> <td class="space">M. Iri, ”Integrability of vector and multivector fields associated with interior point methods for linear programming,”Mathematical Programming (Series B) 52 (1991) 511–525, this issue. · <a href="/0743.90074" class="nowrap">Zbl 0743.90074</a> · <a href="https://doi.org/10.1007/BF01582903" class="nowrap">doi:10.1007/BF01582903</a></td> </tr><tr> <td>[8]</td> <td class="space">Fl. Jarre, ”On the method of analytic centers for solving smooth, convex programs,” in: S. Dolecki, ed.,Optimization, Proceedings of the Fifth French–German Conference. Lecture Notes in Mathematics No. 1405 (Springer, Berlin, 1989) pp. 69–85.</td> </tr><tr> <td>[9]</td> <td class="space">Fl. Jarre, G. Sonnevend and J. Stoer, ”An implementation of the method of analytic centers,” in: A. Bensoussan and J.L. Lions, eds.,Analysis and Optimization of Systems. Lecture Notes in Control and Information Sciences No. 111 (Springer, Berlin, 1988) pp. 297–307.</td> </tr><tr> <td>[10]</td> <td class="space">N. Karmarkar, ”Riemannian geometry underlying interior point methods,” Lecture and Preprint presented at the13th International Symposium on Mathematical Programming, Tokyo (Tokyo, 1988). · <a href="/0725.90058" class="nowrap">Zbl 0725.90058</a></td> </tr><tr> <td>[11]</td> <td class="space">M. Kojima, Sh. Mizuno and A. Yoshise, ”An $O(\sqrt n L)$ iteration potential reduction algorithm for linear complementarity problems,”Mathematical Programming 50 (1991) 331–342. · <a href="/0738.90077" class="nowrap">Zbl 0738.90077</a> · <a href="https://doi.org/10.1007/BF01594942" class="nowrap">doi:10.1007/BF01594942</a></td> </tr><tr> <td>[12]</td> <td class="space">I.J. Lustig, R.E. Marsten and D.E. Shanno, ”Computational experience with a primal-dual interior point method for linear programming,” Technical Report SOR 89-17, School of Engineering and Applied Science, Princeton University (Princeton, NJ, 1989). · <a href="/0731.65049" class="nowrap">Zbl 0731.65049</a></td> </tr><tr> <td>[13]</td> <td class="space">A. Marxen, ”Primal barrier methods for linear programming,” Ph.D. Thesis, Department of Operations Research, Stanford University (Stanford, CA, 1989).</td> </tr><tr> <td>[14]</td> <td class="space">J. Mennicken, ”Implementation of a first order central path following algorithm for solving large linear programs,” Report No. 202, Institut für Angewandte Mathematik und Statistisk, Universität Würzburg (Würzburg, 1990).</td> </tr><tr> <td>[15]</td> <td class="space">S. Mizuno, M.J. Todd and Y. Ye, ”Anticipated behaviour of path-following algorithms for linear programming,” Technical Report No. 878-1989, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1989).</td> </tr><tr> <td>[16]</td> <td class="space">Ju. Nesterov and A. Nemirovskii,Self-Concordant Functions and Polynomial-Time Algorithms for Convex Programming (Central Economical and Mathematical Institute USSR Academy of Science, Moscow, 1989).</td> </tr><tr> <td>[17]</td> <td class="space">C. Roos and J.-Ph. Vial, ”A polynomial method of approximate weighted centers for linear programming,” to appear in:Mathematical Programming (1992). · <a href="/0771.90067" class="nowrap">Zbl 0771.90067</a></td> </tr><tr> <td>[18]</td> <td class="space">G. Sonnevend, ”Application of analytic centers to feedback design for systems with uncertainties,” in: D. Hinrichsen and B. Martensson, eds.,Control of Uncertain Systems (Birkhäuser, Basel, 1989) pp. 271–289.</td> </tr><tr> <td>[19]</td> <td class="space">G. Sonnevend, ”Sequential algorithms of optimal order global error for the uniform recovery of functions with monotonerth derivatives,”Analysis Mathematica 10 (1984) 311–335. · <a href="/0577.65002" class="nowrap">Zbl 0577.65002</a> · <a href="https://doi.org/10.1007/BF01904781" class="nowrap">doi:10.1007/BF01904781</a></td> </tr><tr> <td>[20]</td> <td class="space">G. Sonnevend and J. Stoer, ”Global ellipsoidal approximations and homotopy methods,”Applied Mathematics and Optimization 21 (1989) 139–166. · <a href="/0691.90071" class="nowrap">Zbl 0691.90071</a></td> </tr><tr> <td>[21]</td> <td class="space">G. Sonnevend, J. Stoer and G. Zhao, ”On the complexity of following the central path of linear programs by linear extrapolation,”Methods of Operations Research 62 (1990) 19–31. · <a href="/0726.90056" class="nowrap">Zbl 0726.90056</a></td> </tr><tr> <td>[22]</td> <td class="space">M. Todd, ”Recent developments and new directions in linear programming,” in: M. Iri and K. Tanabe, eds.,Mathematical Programming – Recent Developments and Applications (Kluwer Academic Publishers, Dordrecht–Boston–London, 1989) pp. 109–157. · <a href="/0683.90044" class="nowrap">Zbl 0683.90044</a></td> </tr><tr> <td>[23]</td> <td class="space">C. Witzgall, P.T. Boggs and P.D. Domich, ”On the convergence behavior of trajectories for linear programming,” to appear in:Proceedings of the AMS-IME-SIAM Research Conference on ”Mathematical Developments Arising from Linear Programming Algorithms,” June 26–30, 1988, Bowdoin College (Brunswick, ME). · <a href="/0725.90060" class="nowrap">Zbl 0725.90060</a></td> </tr><tr> <td>[24]</td> <td class="space">Y. Ye, ”A class of potential functions for linear programming,” Preprint, Department of Management Science, The University of Iowa (Iowa City, IA, 1988/1989).</td> </tr><tr> <td>[25]</td> <td class="space">G. Zhao and J. Stoer, ”Estimating the complexity of path following methods for solving linear programs by curvature integrals,” Technical Report No. 225, Institut für Angewandte Mathematik und Statistik, Universität Würzburg (Würzburg, 1990). · <a href="/0768.90056" class="nowrap">Zbl 0768.90056</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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