Hybrid methods to solve absolute value equations in the optimization model

In this paper, we suggest and analyze a hybrid method included iterative method and smoothing-type algorithm for solving the absolute value equations Ax - x =b, where A∈Rn ×n is symmetric matrix, b∈Rn and x∈Rn is unknown. We suggest an iterative method can be viewed as a modification of Gauss-Seidel method for solving the absolute value equations. We also discuss the convergence of the proposed method under suitable conditions. Several examples are given to illustrate the implementation and efficiency of the method. Some open problems are also suggested. This paper, we reformulate the system of absolute value equations as a family of parameterized smooth equations and propose a smoothing Newton method to solve this class of problems. We prove that the method is globally and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate that the method is effective. The preliminary numerical results are reported, which show both methods are effective for solving the absolute value equations Hybrid methods to solve absolute value equations linear problems and non-linear problems in the optimization is useful