Study On Efficient Splitting Iterative Methods For Solving Generalized Absolute Value Equations

Generalized absolute value equation(GAVE)is an important nonlinear nondifferentiable optimization problem.The main research background of GAVE is linear complementarity problem(LCP),which is an optimization problem with wide range of practical applications.Under some certain conditions,the LCP can be transformed into GAVE.In this thesis,we establish two efficient splitting iteration methods for solving large sparse GAVE.The convergence of the new methods are discussed in depth.In addition,the feasibilities and efficiencies of the proposed methods are verified by numerical experiments.This thesis is organized as follows:In Chapter 1,we give the research background,research sources and status of the GAVE,and introduce the related preliminary knowledge.In Chapter 2,by employing out-inner iteration techniques,we use the generalized positive definite and skew-Hermitian splitting(GPSS)iteration as the inner solver of the Picard method,and establish the Picard-GPSS iterative method for solving the GAVE.The convergence conditions of the proposed method are analyzed.Finally,numerical experiments show the efficiency of Picard-GPSS iterative method.In Chapter 3,by utilizing shift splitting the coefficient matrix of the linear part,a shift splitting modified Newton-type(SSMN)iterative method for solving GAVE is proposed.We discuss the convergence conditions of the SSMN iteration method.Furthermore,some sufficient convergence conditions are derived when the coefficient matrix is a symmetric positive definite matrix or an H+-matrix.Two numerical examples indicate that the SSMN iteration method is an effective method to solve the GAVE.In Chapter 4,we summary the research results of this work and point out some problems which may be considered in the future.