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A sub-additive DC approach to the complementarity problem

Author

Listed:
  • L. Abdallah

    (Lebanese University)

  • M. Haddou

    (Univ Rennes)

  • T. Migot

    (University of Guelph)

Abstract

In this article, we study a merit function based on sub-additive functions for solving the non-linear complementarity problem (NCP). This leads to consider an optimization problem that is equivalent to the NCP. In the case of a concave NCP this optimization problem is a Difference of Convex (DC) program and we can therefore use DC Algorithm to locally solve it. We prove that in the case of a concave monotone NCP, it is sufficient to compute a stationary point of the optimization problem to obtain a solution of the complementarity problem. In the case of a general NCP, assuming that a DC decomposition of the complementarity problem is known, we propose a penalization technique to reformulate the optimization problem as a DC program and prove that local minima of this penalized problem are also local minima of the merit problem. Numerical results on linear complementarity problems, absolute value equations and non-linear complementarity problems show that our method is promising.

Suggested Citation

  • L. Abdallah & M. Haddou & T. Migot, 2019. "A sub-additive DC approach to the complementarity problem," Computational Optimization and Applications, Springer, vol. 73(2), pages 509-534, June.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:2:d:10.1007_s10589-019-00078-w
    DOI: 10.1007/s10589-019-00078-w
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    References listed on IDEAS

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    1. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    2. Mounir Haddou & Patrick Maheux, 2014. "Smoothing Methods for Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 711-729, March.
    3. Hoai Le Thi & Tao Pham Dinh, 2011. "On solving Linear Complementarity Problems by DC programming and DCA," Computational Optimization and Applications, Springer, vol. 50(3), pages 507-524, December.
    4. Olvi L. Mangasarian, 2014. "Absolute Value Equation Solution Via Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 870-876, June.
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