P. Tseng et al., EQUIVALENCE OF COMPLEMENTARITY-PROBLEMS TO DIFFERENTIABLE MINIMIZATION - A UNIFIED APPROACH, SIAM journal on optimization, 6(2), 1996, pp. 446-460
We consider two merit functions for a generalized nonlinear complement
arity 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 Solo
dov, 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 poi
nts of this function are the solutions to GNCP. These results extend t
hose of Luo et al. [Math. Oper, Res,, 19 (1994), pp, 880-892] and of Y
amashita and Fukushima [J. Optim. Theory Appl., 84 (1995), pp. 653-663
] on the properties of the implicit Lagrangian.