A new merit function and a successive quadratic programming algorithm for variational inequality problems. (English) Zbl 0853.49011

Taji, Kouichi; Fukushima, Masao

SIAM J. Optim. 6, No. 3, 704-713 (1996).

Summary: Recently, various merit functions for variational inequality problems have been proposed and their properties have been studied. Unfortunately, these functions may not be easy to evaluate unless the constraints of the problem have a relatively simple structure. In this paper, a new merit function for variational inequality problems with general convex constraints is proposed. At each point, the proposed function is defined as an optimal value of a quadratic programming problem whose constraints consist of a linear approximation of the given nonlinear constraints. It is shown that the set of constrained minima of the proposed merit function coincides with the set of solutions to the original variational inequality problem. It is also shown that this function is directionally differentiable in all directions and, under suitable assumptions, any stationary point of the function over the constraint set actually solves the original variational inequality problem. Finally, a descent method for solving the variational inequality problem is proposed and its convergence is proved. The method is closely related to a successive quadratic programming method for solving nonlinear programming problems.

MSC:

49J40	Variational inequalities
90C20	Quadratic programming
90C30	Nonlinear programming

Keywords:

global convergence; variational inequality problems; merit function; convex constraints; successive quadratic programming; nonlinear programming problems