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First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints

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  • Yi Zhang
  • Jia Wu
  • Liwei Zhang

Abstract

This paper is to develop first order necessary optimality conditions for a mathematical program with second-order cone complementarity constraints (MPSCC) which includes the mathematical program with (vector) complementarity constraints (MPCC) as a special case. Like the case of MPCC, Robinson’s constraint qualification fails at every feasible point of MPSCC if we treat the MPSCC as an ordinary optimization problem. Using the formulas of regular and limiting coderivatives and generalized Clarke’s Jacobian of the projection operator onto second-order cones from the literature, we present the S-, M-, C- and A-stationary conditions for a MPSCC problem. Moreover, several constraint qualifications including MPSCC-Abadie CQ, MPSCC-LICQ, MPSCC-MFCQ and MPSCC-GMFCQ are proposed, under which a local minimizer of MPSCC is shown to be a S-, M-, C- or A-stationary point. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Yi Zhang & Jia Wu & Liwei Zhang, 2015. "First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints," Journal of Global Optimization, Springer, vol. 63(2), pages 253-279, October.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:2:p:253-279
    DOI: 10.1007/s10898-015-0295-2
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    2. M.L. Flegel & C. Kanzow, 2005. "Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 595-614, March.
    3. Jia Wu & Liwei Zhang & Yi Zhang, 2013. "A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations," Journal of Global Optimization, Springer, vol. 55(2), pages 359-385, February.
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    Cited by:

    1. April Sagan & Xin Shen & John E. Mitchell, 2020. "Two Relaxation Methods for Rank Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 806-825, September.

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