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Title of Journal: J Optim Theory Appl

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Abbravation: Journal of Optimization Theory and Applications

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Springer US

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DOI

10.1007/s00704-003-0011-x

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1573-2878

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Mathematical Programs with Vanishing Constraints

Authors: A F Izmailov M V Solodov
Publish Date: 2009/03/18
Volume: 142, Issue: 3, Pages: 501-532
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Abstract

We consider a class of optimization problems with switchoff/switchon constraints which is a relatively new problem model The specificity of this model is that it contains constraints that are being imposed switched on at some points of the feasible region while being disregarded switched off at other points This seems to be a potentially useful modeling paradigm that has been shown to be helpful for example in optimal topology design The fact that some constraints “vanish” from the problem at certain points gave rise to the name of mathematical programs with vanishing constraints MPVC It turns out that such problems are usually degenerate at a solution but are structurally different from the related class of mathematical programs with complementarity constraints MPCC In this paper we first discuss some known first and secondorder necessary optimality conditions for MPVC giving new very short and direct justifications We then derive some new special secondorder sufficient optimality conditions for these problems and show that quite remarkably these conditions are actually equivalent to the classical/standard secondorder sufficient conditions in optimization We also provide a sensitivity analysis for MPVC Finally a relaxation method is proposed For this method we analyze constraints regularity and boundedness of the Lagrange multipliers in the relaxed subproblems derive a sufficient condition for local uniqueness of solutions of subproblems and give convergence estimatesResearch of the first author was supported by the Russian Foundation for Basic Research Grants 070100270 070100416 and 070190102Mong and by RF President’s Grant NS934420061 for the support of leading scientific schools The second author was supported in part by CNPq Grants 301508/20054 490200/20052 and 550317/20058 by PRONEXOptimization and by FAPERJ

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