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

• 成果类型: Article
• 署名作者: Ding, Chao; Sun, Defeng; Ye, Jane J.
• 署名单位: Chinese Academy of Sciences; National University of Singapore; National University of Singapore; University of Victoria
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-013-0735-z
• 发表日期: 2014
• 页码: 539-579
• 关键词: optimization problems; global optimization; LARGEST EIGENVALUES; nonconvex nlps; systems; semismoothness; approximation; minimization; nonsmooth; calmness

摘要

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.