AN ITERATIVE ALGORITHM FOR THE VARIATIONAL INEQUALITY PROBLEM

We present a new algorithm for the variational inequality problem, which, given a monotone operator F and a convex set C, consists of finding z is an element of C such that [F(x),x - z] greater than or equal to 0 for all x is an element of C. Our algorithm proceeds as follows. Let P be the orthogonal projection onto C. Given x(k) is an element of C, a positive scalar gamma(k) is found through a finite bracketing procedure, so that, if y(k) = P(x(k) - gamma(k)F(x(k))), the hyperplane normal to F(y(k)) passing through y(k) separates x(k) from the solutions of the problem. x(k+1) is then determined by consecutive orthogonal projections of x(k) onto this hyperplane and onto G. We prove that the sequence so generated converges if and only if the problem has solutions and that, when convergent, its limit is a solution. At variance with other methods, ours requires minimal hypotheses on F, just monotonicity and continuity.