Projective MAOR Iterative Algorithms For Solving Some Classes Of Variational Inequalities

Complementarity problems and two-sided obstacle problems are two basic classes ofvariational inequalities, broadly applied to physics, optimal control theory, engineeringtechnique, tra?c assignment and economic equilibria modeling, etc. So it is significantto establish rapid numerical methods to solve these problems. Over the past decades,lots of e?cient algorithms have been developed. In this paper, we discuss and analyzeprojective modified accelerated overrelaxation (MAOR) algorithm for solving implicitcomplementarity problems, implicit two-sided obstacle problems and implicit two-sidedobstacle problems with nonlinear source terms.MAOR algorithm was firstly developed to solve linear systems, and it includes manyclassic iterative methods. MAOR algorithm may be better than other iterates, for ithas more relaxation factors, which can be chosen properly to accelerate the iterationprocess. In this paper, we extend MAOR algorithm to solve implicit complementarityproblems with L?matrix. We establish a projective MAOR algorithm and show that anyaccumulation point of the iteration generated by the algorithm solves the implicit comple-mentarity problem. Moreover, when the coe?cient matrix A involved in the problem is anM?matrix, the sequence monotonically converges to the solution. We also use projectiveMAOR algorithm to solve implicit two-sided obstacle problems. The algorithms based onthe set of lower solutions and upper solutions respectively also possess the monotone con-vergence. Moreover, we discuss projective MAOR algorithm to solve implicit two-sidedproblems with nonlinear source terms, and similar theoretical results are obtained. Inthe last section of this paper, numerical experiment tests are presented, which verify theresults we obtained.