A self-adaptive method for solving general mixed variational inequalities

Abstract

The general mixed variational inequality containing a nonlinear term ϕ is a useful and an important generalization of variational inequalities. The projection method cannot be applied to solve this problem due to the presence of the nonlinear term. To overcome this disadvantage, Noor [M.A. Noor, Pseudomonotone general mixed variational inequalities, Appl. Math. Comput. 141 (2003) 529-540] used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities. In this paper, we present a new self-adaptive iterative method which can be viewed as a refinement and improvement of the method of Noor. Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given.  2004 Elsevier Inc. All rights reserved.

Inexact proximal point method for general variational inequalities

“…In order to verify the theoretical assertions, we consider the following problems: where A is an n × n matrix, Π is a simple closed convex set in R n , 0 < p ∈ R n is a parameter vector. It has been shown [2] that solving problem (6.1) is equivalent to find a pair (u * , y * ), such that βf u * = A T y * (6.2) and g u * ∈ Π, g(v) − g u * T y * 0 ∀g(v) ∈ Π, (6.3)…”

“…Since in many cases solving problem (1.5) exactly is either impossible or as difficult as solving the original problem, various forms of inexact PPA have been developed, see [2,11,[19][20][21][22][23][24][25][26][27][28] and references therein.…”

Some iterative schemes for general mixed variational inequalities

“…It is well known that a large class of problems arising in industry, ecology, finance, economics, transportation, network analysis and optimization can be formulated and studied in the framework of (2.1) and (2.7), see [1][2][3] and the references therein.…”

“…This alternative equivalent form plays a crucial role in suggesting three-step iterative schemes for solving variational inequalities. It is now well-known [1,2,31] that the three-step iterative methods for solving the variational inequalities perform better numerically than two-step and one-step method. This fact motivates us to consider the three-step iterative schemes.…”

“…Clearly Noor iterations include Mann (onestep) and Ishikawa (two-step) iterations as special cases. It has been shown in [1,2,31] that three-step iterative methods are numerically more efficient than one-step and two-step iterative methods for solving variational inequalities and the related optimization problems.…”