A sub-additive DC approach to the complementarity problem

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.

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Acknowledgements

The authors would like to thank anonymous referees for their helpful remarks and comments.

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Correspondence to T. Migot.

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Abdallah, L., Haddou, M. & Migot, T. A sub-additive DC approach to the complementarity problem. Comput Optim Appl 73, 509–534 (2019). https://doi.org/10.1007/s10589-019-00078-w

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Keywords

  • Complementarity problem
  • Difference of convex
  • Merit function
  • DC algorithm

Mathematics Subject Classification

  • 90C59
  • 90C30
  • 90C33
  • 65K05
  • 49M20