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Mathematical Programming

, Volume 147, Issue 1–2, pp 539–579 | Cite as

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

  • Chao Ding
  • Defeng Sun
  • Jane J. YeEmail author
Full Length Paper Series A

Abstract

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.

Keywords

Mathematical program with semidefinite cone complementarity constraints Necessary optimality conditions Constraint qualifications S-stationary conditions M-stationary conditions C-stationary conditions 

Mathematics Subject Classification

49K10 49J52 90C30 90C22 90C33 

Notes

Acknowledgments

The authors are grateful to the anonymous referees for their constructive suggestions and comments which helped to improve the presentation of the materials in this paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.National Center for Mathematics and Interdisciplinary SciencesChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and Risk Management InstituteNational University of SingaporeSingaporeRepublic of Singapore
  3. 3.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

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