A Merit Function for Variational Inequalities Applied to EquilibriumProblems
Computational Economics
Merit Functions for Complementarity and Related Problems: A Survey
Computational Optimization and Applications
Global convergence on an active set SQP for inequality constrained optimization
Journal of Computational and Applied Mathematics
On Nash---Cournot oligopolistic market equilibrium models with concave cost functions
Journal of Global Optimization
Global convergence on an active set SQP for inequality constrained optimization
Journal of Computational and Applied Mathematics
Gap functions and penalization for solving equilibrium problems with nonlinear constraints
Computational Optimization and Applications
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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.