Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets

A smoothing homotopy method for solving variational inequalities

“…In the section, we implement the smoothing homotopy method based on the homotopy (5) as well as combined homotopy method in [15,16] to solve the VIP(X, F) in MATLAB. For both methods, Algorithm 3.1 is used to numerically trace the homotopy path.…”

“…For both methods, Algorithm 3.1 is used to numerically trace the homotopy path. In the given algorithm, for the smoothing homotopy method (5), we take S(w, µ) = (1 − µ)Φ µ (x) + µ(x − x (0) ), T (w) = g(x), w = x and the smoothing function p(x, µ) in the homotopy (5) is chosen to be the Chen-Harker-Kanzow-Smale smoothing function [7,29,30]; For the combined homotopy method [15,16], we take…”

“…We deal with the smoothing-nonsmooth equation reformulation of the VIP(X, F) instead of the method as in [14,16] by the homotopy method, where the smooth approximation function is an arbitrary one which satisfies certain conditions. Under only a certain condition on the defining mapping F, for the starting point chosen almost everywhere in R n , existence and convergence of the homotopy pathway are proved.…”

“…Later, it was developed to be a powerful tool to solve not only the convex programming problems [23] but also the nonconvex problems(see, e.g., [24,25]). For the nonmonotone variational inequality problem, Lin et al [14] and Xu et al [15,16] utilized the combined homotopy method to solve its KKT system in a bounded set and unbounded set X which is a closed convex set with nonempty interior and represented by several inequalities. Existence and convergence of homotopy pathway were proved without monotonicity of F. However, the combined homotopy method to solve the KKT system increases the number of the variables which can bring more trouble in dealing with the problems.…”

“…Therefore, to improve the locally convergent nature of most algorithms, recent research has been devoted to creating globally convergent methods. These solution methods for the variational inequality problems can be classified into three categories: (1) transforming into nonsmooth equations (or, fixed-point problems) and then being solved by semismooth Newton-type methods, smoothing Newton methods, continuation method or projective methods (see, e.g., [4][5][6][7]2,[8][9][10]); (2) reformulating as optimization problems and then being solved by some algorithms for optimization problems (see, e.g., [11][12][13]); (3) solving KKT systems of the variational inequality problems similarly with KKT system of constrained optimization (homotopy methods, e.g., [14][15][16]). However, the convergence of many algorithms were established when the mapping F is assumed to have some monotonicity.…”