A GENERAL DESCENT FRAMEWORK FOR THE MONOTONE VARIATIONAL INEQUALITY PROBLEM

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP(v) which consists in finding a solution v* is-an-element-of OMEGA(v) satisfying s(v*)T(v - v* ) greater-than-or-equal-to 0, for all v is-an-element-of OMEGA(v). This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.