The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

  • Authors:

  • George Isac;George Xian-Zhi Yuan

  • Affiliations:

  • Department of Mathematics and Computers, Royal Military College of Canada, Kingston, Ontario, Canada K7K 7B4/;Department of Mathematics, The University of Queensland, Brisbane, QLD, Australia 4072. E-mail: xzy@maths.uq.edu.au Corresponding author.

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2000

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Abstract

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.