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A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

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  • Jia Wu
  • Liwei Zhang
  • Yi Zhang

Abstract

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [ 40 ] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:55:y:2013:i:2:p:359-385
    DOI: 10.1007/s10898-012-9880-9
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    References listed on IDEAS

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    1. G.H. Lin & M. Fukushima, 2003. "Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 67-80, July.
    2. Gemayqzel Bouza & Georg Still, 2007. "Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 467-483, May.
    3. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    4. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
    5. Jia Chen & Yeol Cho & Jong Kim & Jun Li, 2011. "Multiobjective optimization problems with modified objective functions and cone constraints and applications," Journal of Global Optimization, Springer, vol. 49(1), pages 137-147, January.
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    Cited by:

    1. 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.
    2. Xide Zhu & Jin Zhang & Jinchuan Zhou & Xinmin Yang, 2019. "Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 521-540, May.

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