A Gradient Algorithm For Solving The Absolute Value Equation

Absolute value equation(AVE)Ax-|x| = b is a special class of nonlinear equations and it is NP-hard.The absolute value equation is derived from the interval problem and is now applied to many practical problems,such as the feasibility problem of knapsack,the location problem and the problem of unsupervised and semi supervised classification.On the other hand,the absolute value equation is equivalent to the linear complementarity problem,and the optimization problems such as the traditional linear programming,the two programming and the dual matrix can be transformed into the linear complementarity problem.Therefore,the study of the absolute value equation provides a new solution for many mathematical programming problems.In summary,the absolute value equation of study has important significance.This paper focuses on the solution of AVE under the conditions that it has solu-tion.Firstly,under the condition of that the coefficient matrix A is symmetric positive definite,we give the PRP conjugate gradient method for solving the absolute value equation,and analyze its convergence,and the numerical experiment also shows the effectiveness of the algorithm.In second,under the conditions of that the matrix A is nonsymmetric and positive definite,the absolute value equation will be transformed into linear equations whose coefficient matrix is symmetric positive definite,and the equations will be solved by preconditioning conjugate gradient method,so as to obtain the solution of the original problem.Numerical experiment results show the effective-ness of the new method.