Homotopy Methods for Solving Variational Inequalities in Unbounded Sets

In this paper, for solving the finite-dimensional variational inequality problem ðx x ÞFðx ÞP0; 8x 2 X; where F is a C ðr > 1Þ mapping from X to R, X 1⁄4 fx 2 R : gðxÞO0g is nonempty (not necessarily bounded) and gðxÞ :R ! R is a convex Crþ1 mapping, a homotopy method is presented. Under various conditions, existence and convergence of a smooth homotopy path from almost any interior initial point in X to a solution of the variational inequality problem is proven. It leads to an implementable and globally convergent algorithm and gives a new and constructive proof of existence of solution.