Inexact proximal point method for general variational inequalities

Abstract

In this paper, we suggest and analyze a new inexact proximal point method for solving general variational inequalities, which can be considered as an implicit predictor-corrector method. An easily measurable error term is proposed with further relaxed error bound and an optimal step length is obtained by maximizing the profit-function and is dependent on the previous points. Our results include several known and new techniques for solving variational inequalities and related optimization problems. Results obtained in this paper can be viewed as an important improvement and refinement of the previously known results. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.

On convergence criteria of generalized proximal point algorithms

“…In recent years, much attention has been given to develop several iterative algorithms including the proximal point algorithms for solving the variational inclusions involving the maximal monotone operators, see [1,2,4,[6][7][8][9][10][11][12][13][14][15] and the references therein. Inspired and motivated by the recent research going on in this area, we analyze some generalized proximal point algorithms.…”

An inexact operator splitting method for general mixed variational inequalities

“…GVI has found many ecient applications in various application domains. We refer the readers to [3,13,19,22] for some review papers.…”

“…Based on the work of He et al [15], Li and Bnouhachem [18] proposed another inexact method for solving (3), where the new iteration is generated via the following recursion:…”

Algorithmic Approach to the Split Problems

“…for all Ψ( ) ∈ C. We use VI(A, Ψ) to denote the set of solutions of (2). Variational inequality problems have important applications in many fields such as elasticity, optimization, economics, transportation, and structural analysis, and various numerical methods have been studied by many researchers; see, for instance, [9][10][11][12][13][14][15][16][17].…”