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

C. Ding, Defeng Sun, J.J. Ye

Research output: Journal article publicationJournal articleAcademic researchpeer-review

34 Citations (Scopus)


© 2013, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society. 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.
Original languageEnglish
Pages (from-to)539-579
Number of pages41
JournalMathematical Programming
Issue number1-2
Publication statusPublished - 1 Jan 2013
Externally publishedYes


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

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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