Equivalence of Complementarity Problems to Differentiable Minimization: A Unified Approach

Abstract

We consider two merit functions for a generalized nonlinear complementarity problem (GNCP) based on quadratic regularization of the standard linearized gap function. The first extends Fukushima’s merit function for variational inequality problems [Fukushima, Math. Programming, 53 (1992), pp. 99–110] and the second extends Mangasarian and Solodov’s implicit Lagrangian for complementarity problems [Mangasarian and Solodov, Math. Programming, 62 (1993), pp. 277–297]. We show, among other things, that the second merit function is in the order of the natural residual squared and we give conditions under which the stationary points of this function are the solutions to GNCP. These results extend those of Luo et al. [Math. Oper. Res., 19 (1994), pp. 880–892] and of Yamashita and Fukushima [J. Optim. Theory Appl., 84 (1995), pp. 653–663] on the properties of the implicit Lagrangian.