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Growth behavior of a class of merit functions for the nonlinear complementarity problem

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Abstract

When the nonlinear complementarity problem is reformulated as that of finding the zero of a self-mapping, the norm of the selfmapping serves naturally as a merit function for the problem. We study the growth behavior of such a merit function. In particular, we show that, for the linear complementarity problem, whether the merit function is coercive is intimately related to whether the underlying matrix is aP-matrix or a nondegenerate matrix or anR o-matrix. We also show that, for the more popular choices of the merit function, the merit function is bounded below by the norm of the natural residual raised to a positive integral power. Thus, if the norm of the natural residual has positive order of growth, then so does the merit function.

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Additional information

This work was partially supported by the National Science Foundation Grant No. CCR-93-11621.

The author thanks Dr. Christian Kanzow for his many helpful comments on a preliminary version of this paper. He also thanks the referees for their helpful suggestions.

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Tseng, P. Growth behavior of a class of merit functions for the nonlinear complementarity problem. J Optim Theory Appl 89, 17–37 (1996). https://doi.org/10.1007/BF02192639

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Key Words

  • Complementarity problems
  • merit functions
  • error bounds
  • natural residuals
  • coercive functions