Optimality of orders one to three and beyond: Characterization and evaluation complexity in constrained nonconvex optimization

Coralia Cartis, N. I. M. Gould, Philippe Toint

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Résumé

Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second- and third-order criticality and its evaluation complexity is analyzed as a function of the choice (among existing methods) of an inner algorithm for solving subproblems in each of the two phases. The relation between high-order criticality and penalization techniques is finally considered, showing that standard algorithmic approaches will fail if approximate constrained high-order critical points are sought.

langue originaleAnglais
Pages (de - à)68-94
Nombre de pages32
journalJournal of Complexity
Volume53
Les DOIs
Etat de la publicationPublié - 10 août 2019

Financement

The authors would like to thank Oliver Stein for suggesting reference [41] . The work of the second author was supported by EPSRC, United Kingdom grant EP/M025179/1 . The third author acknowledges the support provided by the Belgian Fund for Scientific Research (FNRS) , the Leverhulme Trust (UK) , Balliol College (Oxford, UK) , the Department of Applied Mathematics of the Hong Kong Polytechnic University , ENSEEIHT (Toulouse, France) and INDAM (Florence, Italy) . Thanks are also due to a thoughtful referee whose patience and perceptive comments have helped to significantly improve the manuscript. Appendix

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