A projection descent method for solving variational inequalities

Journal of Optimization Theory and Applications, 2013

In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied.

Acta Mathematica Sinica, English Series, 2006

In this paper we consider the general variational inequality GVI(F, g, C) where F and g are mappings from a Hilbert space into itself and C is the fixed points set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI(F, g, C). Strong convergence results are established and applications to constrained generalized pseudoinverse are included.

2002

In this paper, we consider and analyze a new class of projection methods for solving pseudomonotone general variational inequalities using the Wiener-Hopf equations technique. The modified methods converge for pseudomonotone operators. Our proof of convergence is very simple as compared with other methods. The proposed methods include several known methods as special cases.  2002 Elsevier Science (USA)

Applicable Analysis, 2018

In this paper, a new projection-type algorithm for solving multi-valued mixed variational inequalities without monotonicity is presented. Under some suitable assumptions, it is showed that the sequence generated by the proposed algorithm converges globally to a solution of the multivalued mixed variational inequality considered. The algorithm exploited in this paper is based on the generalized f-projection operator due to Wu and Huang [The generalized f-projection operator with an application. Bull Austral Math Soc. 2006;73:307-317] rather than the well-known resolvent operator. Preliminary computational experience is also reported. The results presented in this paper generalize and improve some known results given in the literature.

Applied Mathematics and Computation, 2014

Based on the logarithmic-quadratic proximal method, a descent alternating direction method is introduced for structured variational inequalities in the paper. The proposal method generates the new iteration by searching the optimal step size along the integrated descent direction from two descent directions. Global convergence of the proposal method is proved under certain assumptions. To illustrate the proposal method and demonstrate its efficiency, some applications and their numerical results are also provided.

Applied Mathematics and Computation, 2007

In this paper, we introduce and consider a new system of variational inequalities involving two different operators. Using the projection technique, we suggest and analyze new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving the single operator, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

Mathematical Programming, 1993

We present a framework for descent algorithms that solve the monotone variational inequality problem V IP v which consists in nding a solution v 2 v which satis es s(v) T (u?v) 0, for all u 2 v. This uni ed framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this uni ed algorithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.