Document Zbl 1408.49027

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Program.</a> <a href="/?q=in%3A427608" title="Articles in this Issue">173, No. 1-2 (A), 221-249 (2019)</a>. </div> <div class="abstract">Summary: We propose a complete-search algorithm for solving a class of nonconvex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an $\varepsilon $-suboptimal global solution within finite run-time for any given termination tolerance $\varepsilon > 0$. Finally, we illustrate these results for a problem of calculus of variations.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7014732">Cited in <strong>5</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 49M30 </td> <td class="space"> Other numerical methods in calculus of variations (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K10" title="MSC2020">65K10</a> </td> <td class="space"> Numerical optimization and variational techniques </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 class="mono"> 93B40 </td> <td class="space"> Computational methods in systems theory (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ainfinite-dimensional+optimization">infinite-dimensional optimization</a>; <a href="/?q=ut%3Acomplete+search">complete search</a>; <a href="/?q=ut%3Abranch-and-lift">branch-and-lift</a>; <a href="/?q=ut%3Aconvergence+analysis">convergence analysis</a>; <a href="/?q=ut%3Acomplexity+analysis">complexity analysis</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4764">ACADO</a>; <a href="/software/5897">Chebfun</a>; <a href="/software/6274">LiftOpt</a>; <a href="/software/9241">ANTIGONE</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 1408.49027" data-ciurl="/ci/07014732" data-biburl="/bibtex/07014732.bib" data-amsurl="/amsrefs/07014732.bib" data-xmlurl="/xml/07014732.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07014732.pdf" title="Zbl 1408.49027 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/s10107-017-1215-7" aria-label="DOI for “Global optimization in Hilbert space”" title="10.1007/s10107-017-1215-7">DOI</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">Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Translated by N. Kemmer. Hafner Publishing Co., New York (1965) · <a href="/0135.33803" class="nowrap">Zbl 0135.33803</a></td> </tr><tr> <td>[2]</td> <td class="space">Albersmeyer, J., Diehl, M.: The lifted Newton method and its application in optimization. SIAM J. Optim. 20(3), 1655-1684 (2010) · <a href="/1198.90396" class="nowrap">Zbl 1198.90396</a></td> </tr><tr> <td>[3]</td> <td class="space">Anderson, E.J., Nash, P.: Linear Programming in Infinite-Dimensional Spaces. Wiley, Hoboken (1987) · <a href="/0632.90038" class="nowrap">Zbl 0632.90038</a></td> </tr><tr> <td>[4]</td> <td class="space">Bampou, D., Kuhn, D.: Polynomial approximations for continuous linear programs. SIAM J. Optim. 22(2), 628-648 (2012) · <a href="/1246.93059" class="nowrap">Zbl 1246.93059</a></td> </tr><tr> <td>[5]</td> <td class="space">Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods, and Applications. Springer, Berlin (2004) · <a href="/1059.74001" class="nowrap">Zbl 1059.74001</a></td> </tr><tr> <td>[6]</td> <td class="space">Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. Advances in Design and Control Series, 2nd edn. SIAM, Philadelphia (2010) · <a href="/1189.49001" class="nowrap">Zbl 1189.49001</a></td> </tr><tr> <td>[7]</td> <td class="space">Biegler, L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243-248 (1984)</td> </tr><tr> <td>[8]</td> <td class="space">Bock, H.G., Plitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings 9th IFAC World Congress Budapest, pp. 243-247. Pergamon Press, Oxford (1984)</td> </tr><tr> <td>[9]</td> <td class="space">Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Glob. Optim. 52(1), 1-28 (2012) · <a href="/1257.90077" class="nowrap">Zbl 1257.90077</a></td> </tr><tr> <td>[10]</td> <td class="space">Bompadre, A., Mitsos, A., Chachuat, B.: Convergence analysis of Taylor and McCormick-Taylor models. J. Glob. Optim. 57(1), 75-114 (2013) · <a href="/1295.90052" class="nowrap">Zbl 1295.90052</a></td> </tr><tr> <td>[11]</td> <td class="space">Boyd, S., Vandenberghe, L.: Convex Optimization. University Press, Cambridge (2004) · <a href="/1058.90049" class="nowrap">Zbl 1058.90049</a></td> </tr><tr> <td>[12]</td> <td class="space">Bryson, A.E., Ho, Y.: Applied Optimal Control. Hemisphere, Washington (1975)</td> </tr><tr> <td>[13]</td> <td class="space">Buie, R., Abrham, J.: Numerical solutions to continuous linear programming problems. Z. Oper. Res. 17(3), 107-117 (1973) · <a href="/0264.90027" class="nowrap">Zbl 0264.90027</a></td> </tr><tr> <td>[14]</td> <td class="space">Devolder, O.; Glineur, F.; Nesterov, Y.; Diehl, M. (ed.); Glineur, F. (ed.); Jarlebring, E. (ed.); Michiels, W. (ed.), Solving infinite-dimensional optimization problems by polynomial approximation, 31-40 (2010), Berlin Heidelberg</td> </tr><tr> <td>[15]</td> <td class="space">Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, Berlin (1987) · <a href="/0666.41001" class="nowrap">Zbl 0666.41001</a></td> </tr><tr> <td>[16]</td> <td class="space">Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications, Oxford (2014)</td> </tr><tr> <td>[17]</td> <td class="space">Floudas, C.A.: Deterministic Global Optimization: Theory, Methods, and Applications. Kluwer, Dordrecht (1999)</td> </tr><tr> <td>[18]</td> <td class="space">Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115-1145 (1995) · <a href="/0885.68088" class="nowrap">Zbl 0885.68088</a></td> </tr><tr> <td>[19]</td> <td class="space">Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644-668 (1997) · <a href="/0885.42003" class="nowrap">Zbl 0885.42003</a></td> </tr><tr> <td>[20]</td> <td class="space">Henrion, D., Tarbouriech, S., Arzelier, D.: LMI approximations for the radius of the intersection of ellipsoids: a survey. J. Optim. Theory Appl. 108(1), 1-28 (2001) · <a href="/1033.93058" class="nowrap">Zbl 1033.93058</a></td> </tr><tr> <td>[21]</td> <td class="space">Henrion, D., Korda, M.: Convex computation of the region of attraction of polynomial control systems. IEEE Trans. Autom. Control 59(2), 297-312 (2014) · <a href="/1360.93601" class="nowrap">Zbl 1360.93601</a></td> </tr><tr> <td>[22]</td> <td class="space">Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2009) · <a href="/1167.49001" class="nowrap">Zbl 1167.49001</a></td> </tr><tr> <td>[23]</td> <td class="space">Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin, Germany (1996) · <a href="/0867.90105" class="nowrap">Zbl 0867.90105</a></td> </tr><tr> <td>[24]</td> <td class="space">Houska, B., Ferreau, H.J., Diehl, M.: ACADO toolkit – an open-source framework for automatic control and dynamic optimization. Optim. Control Appl. Methods 32, 298-312 (2011) · <a href="/1218.49002" class="nowrap">Zbl 1218.49002</a></td> </tr><tr> <td>[25]</td> <td class="space">Houska, B., Chachuat, B.: Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control. J. Optim. Theory Appl. 162(1), 208-248 (2014) · <a href="/1307.49034" class="nowrap">Zbl 1307.49034</a></td> </tr><tr> <td>[26]</td> <td class="space">Houska, B., Villanueva, M.E., Chachuat, B.: Stable set-valued integration of nonlinear dynamic systems using affine set parameterizations. SIAM J. Numer. Anal. 53(5), 2307-2328 (2015) · <a href="/1326.65088" class="nowrap">Zbl 1326.65088</a></td> </tr><tr> <td>[27]</td> <td class="space">Jackson, D.: The Theory of Approximation, vol. XI. AMS Colloquium Publication, New York (1930) · <a href="/56.0936.01" class="nowrap">JFM 56.0936.01</a></td> </tr><tr> <td>[28]</td> <td class="space">Katznelson, Y.: An Introduction to Harmonic Analysis, 2nd edn. Dover Publications, New York (1976) · <a href="/0352.43001" class="nowrap">Zbl 0352.43001</a></td> </tr><tr> <td>[29]</td> <td class="space">Korda, M., Henrion, D., Jones, C.N.: Convex computation of the maximum controlled invariant set for polynomial control systems. SIAM J. Control Optim. 52(5), 2944-2969 (2014) · <a href="/1311.49073" class="nowrap">Zbl 1311.49073</a></td> </tr><tr> <td>[30]</td> <td class="space">Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)</td> </tr><tr> <td>[31]</td> <td class="space">Lin, Y., Stadtherr, M.A.: Validated solutions of initial value problems for parametric ODEs. Appl. Numer. Math. 57(10), 1145-1162 (2007) · <a href="/1121.65084" class="nowrap">Zbl 1121.65084</a></td> </tr><tr> <td>[32]</td> <td class="space">Luo, X., Bertsimas, D.: A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Optim. 37, 177-210 (1998) · <a href="/0921.49023" class="nowrap">Zbl 0921.49023</a></td> </tr><tr> <td>[33]</td> <td class="space">McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Program. 10, 147-175 (1976) · <a href="/0349.90100" class="nowrap">Zbl 0349.90100</a></td> </tr><tr> <td>[34]</td> <td class="space">Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2-3), 503-526 (2014) · <a href="/1301.90063" class="nowrap">Zbl 1301.90063</a></td> </tr><tr> <td>[35]</td> <td class="space">Mitsos, A., Chachuat, B., Barton, P.I.: McCormick based relaxations of algorithms. SIAM J. Optim. 20, 573-601 (2009) · <a href="/1192.65083" class="nowrap">Zbl 1192.65083</a></td> </tr><tr> <td>[36]</td> <td class="space">Moore, R.E.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1979) · <a href="/0417.65022" class="nowrap">Zbl 0417.65022</a></td> </tr><tr> <td>[37]</td> <td class="space">Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)</td> </tr><tr> <td>[38]</td> <td class="space">Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45, 236-262 (2007) · <a href="/1141.65056" class="nowrap">Zbl 1141.65056</a></td> </tr><tr> <td>[39]</td> <td class="space">Nemirovski, A., Roos, C., Terlaky, T.: On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. 86(3), 463-473 (1999) · <a href="/0944.90056" class="nowrap">Zbl 0944.90056</a></td> </tr><tr> <td>[40]</td> <td class="space">Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Methods in Convex Programming. SIAM, Philadelphia (1994) · <a href="/0824.90112" class="nowrap">Zbl 0824.90112</a></td> </tr><tr> <td>[41]</td> <td class="space">Nesterov, Y.: Semidefinite relaxation and non-convex quadratic optimization. Optim. Methods Softw. 12, 1-20 (1997)</td> </tr><tr> <td>[42]</td> <td class="space">Nesterov, Y.; Frenk, H. (ed.); Roos, K. (ed.); Terlaky, T. (ed.); Zhang, S. (ed.), Squared functional systems and optimization problems, 405-440 (2000), Dordrecht · <a href="/0958.90090" class="nowrap">Zbl 0958.90090</a></td> </tr><tr> <td>[43]</td> <td class="space">Neumaier, A.: Taylor forms—use and limits. Reliab. Comput. 9(1), 43-79 (2002) · <a href="/1071.65070" class="nowrap">Zbl 1071.65070</a></td> </tr><tr> <td>[44]</td> <td class="space">Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13, 271-369 (2004) · <a href="/1113.90124" class="nowrap">Zbl 1113.90124</a></td> </tr><tr> <td>[45]</td> <td class="space">Parrilo, P.A.: Polynomial games and sum of squares optimization. In: Proceedings of the 45th IEEE Conference on Decision & Control, pp. 2855-2860. San Diego (CA) (2006)</td> </tr><tr> <td>[46]</td> <td class="space">Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962) · <a href="/0102.32001" class="nowrap">Zbl 0102.32001</a></td> </tr><tr> <td>[47]</td> <td class="space">Rajyaguru, J., Villanueva, M.E., Houska, B., Chachuat, B.: Chebyshev model arithmetic for factorable functions. J. Glob. Optim. 68(2), 413-438 (2017) · <a href="/1386.90119" class="nowrap">Zbl 1386.90119</a></td> </tr><tr> <td>[48]</td> <td class="space">Saff, E.B., Totik, V.: Polynomial approximation of piecewise analytic functions. J. Lond. Math. Soc. 39(2), 487-498 (1989) · <a href="/0683.41011" class="nowrap">Zbl 0683.41011</a></td> </tr><tr> <td>[49]</td> <td class="space">Sahinidis, N.V.: A general purpose global optimization software package. J. Glob. Optim. 8(2), 201-205 (1996) · <a href="/0856.90104" class="nowrap">Zbl 0856.90104</a></td> </tr><tr> <td>[50]</td> <td class="space">Scott, J.K., Chachuat, B., Barton, P.I.: Nonlinear convex and concave relaxations for the solutions of parametric ODEs. Optim. Control Appl. Methods 34(2), 145-163 (2013) · <a href="/1273.93089" class="nowrap">Zbl 1273.93089</a></td> </tr><tr> <td>[51]</td> <td class="space">von Stryk, O., Bulirsch, R.: Direct and indirect methods for trajectory optimization. Ann. Oper. Res. 37, 357-373 (1992) · <a href="/0784.49023" class="nowrap">Zbl 0784.49023</a></td> </tr><tr> <td>[52]</td> <td class="space">Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225-249 (2005) · <a href="/1099.90047" class="nowrap">Zbl 1099.90047</a></td> </tr><tr> <td>[53]</td> <td class="space">Villanueva, M.E., Houska, B., Chachuat, B.: Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs. J. Glob. Optim. 62(3), 575-613 (2015) · <a href="/1320.49013" class="nowrap">Zbl 1320.49013</a></td> </tr><tr> <td>[54]</td> <td class="space">Vinter, R.: Optimal Control. Springer, Berlin (2010) · <a href="/1215.49002" class="nowrap">Zbl 1215.49002</a></td> </tr><tr> <td>[55]</td> <td class="space">Wang, H., Xiang, S.: On the convergence rates of Legendre approximation. Math. Comput. 81(278), 861-877 (2012) · <a href="/1242.41016" class="nowrap">Zbl 1242.41016</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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