A Variable-Metric Non-monotone Line Search Method for Mixed Variational Inequalities and Equilibrium Problems

Abstract: We propose a scaled gap-function method for variational inequalities, mixed variational inequalities, and equilibrium problems over a closed convex set. The method drives a gap function to zero by combining a variable-metric scaled projection step with a modified non-monotone Armijo line search. The construction rests on a structural identity: the scaled projection step is exactly the maximizer that defines the Fukushima regularized gap, so the search direction is simultaneously the algorithmic step and the gap-defining direction. For variational and mixed variational inequalities with a strongly monotone, Lipschitz operator we establish global convergence and an R-linear rate, under a fixed or a controlled-change variable metric. The rate follows from the strong-monotonicity contraction and the resulting explicit gap error bound. For equilibrium problems we obtain global convergence and a gap error bound (Mastroeni's gap). Numerical experiments on controlled problems confirm the convergence guarantees and the predicted rate, and compare the method with standard extragradient and gap-descent methods. The combination of variable-metric scaling, a modified non-monotone line search, and gap-function descent appears to be new for these problem classes.