Homotopy Methods for Solving Variational Inequalities in Unbounded Sets

AbstractIn this paper, for solving the finite-dimensional variational inequality problem $$(x-x*)^{T} F(x*)\geq 0, \quad \forall x\in X,$$where F is a $$C^r (r gt; 1)$$ mapping from X to Rn, X = $$ { x \in R^{n} : g(x) leq; 0}$$ is nonempty (not necessarily bounded) and $${\it g}({\it x}): R^{n} \rightarrow R^{m}$$ is a convex Cr+1 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.