A smoothing homotopy method for solving variational inequalities

作者：Fan Xiaona^*; Yu Bo

来源：Nonlinear Analysis-Theory Methods & Applications, 2009, 70(1): 211-219.

DOI：10.1016/j.na.2007.11.047

• Variational inequality
• Smoothing method
• Homotopy method
• Global convergence

In this paper, a new homotopy method for solving the variational inequality problem VIP(X, F): find y* is an element of X such that (y - y*)(T) F(y*) >= 0, for all y is an element of X, where X is a nonempty closed convex subset of R(n) and F : R(n) --> R(n) is a continuously differentiable mapping, is proposed. The homolopy equation is constructed based on the smooth approximation to Robinson';s normal equation of variational inequality problem, where the smooth approximation function p(x, A) of the projection function Pi(X)(x) is an arbitrary one such that for any mu > 0 and x is an element of R(n), p(x, mu) is an element of int X. Under a weak condition on the defining mapping F, which is needed for the existence of a solution to VIP(X, F), for the starting point chosen almost everywhere in R(n), existence and convergence of a smooth homotopy pathway to a solution of VIP(X, F) are proved. Several numerical experiments indicate that the method is efficient.

• 出版日期2009-1-1
• 单位大连理工大学