Abstract
In this paper, we transform the problem of solving the absolute value equations (AVEs)Ax−x=bwith singular values ofAgreater than 1 into the problem of finding the root of the system of nonlinear equation and propose a three-step algorithm for solving the system of nonlinear equation. The proposed method has the global linear convergence and the local quadratic convergence. Numerical examples show that this algorithm has high accuracy and fast convergence speed for solving the system of nonlinear equations.
Highlights
In the last few decades, the system of absolute value equations has been recognized as a kind of NP-hard and nondifferentiable problem, which can be equivalent to many mathematical problems, such as generalized linear complementarity problem, bilinear programming problem [1], knapsack feasibility problem [2], traveling agent problem [3], etc. . .compared with the above problems, it has the characteristics of simple structure and easy solution, which attracts much attention. is paper mainly designed a three-step iterative algorithm to solve the absolute value equations (AVEs) of the following type: Ax − |x| b, (1)
In [4], some theoretical conclusions about the solutions of AVEs (1) are given, and the sufficient conditions for the existence of unique solutions, nonnegative solutions, 2n solutions, and no solutions are given for the first time
Considering the nondifferentiation of absolute value equation, taking the advantages of the above algorithms and overcoming their disadvantages, a three-step iterative algorithm is designed for effectively solving AVEs (1) in this paper
Summary
In the last few decades, the system of absolute value equations has been recognized as a kind of NP-hard and nondifferentiable problem, which can be equivalent to many mathematical problems, such as generalized linear complementarity problem, bilinear programming problem [1], knapsack feasibility problem [2], traveling agent problem [3], etc. . .compared with the above problems, it has the characteristics of simple structure and easy solution, which attracts much attention. is paper mainly designed a three-step iterative algorithm to solve the absolute value equations (AVEs) of the following type: Ax − |x| b,. Is paper mainly designed a three-step iterative algorithm to solve the absolute value equations (AVEs) of the following type: Ax − |x| b,. 100 AVEs (1) of 1000 dimensions generated randomly are used to test effectiveness of this method, and the accuracy is up to 10− 6 On this basis, an improved generalized Newton algorithm is proposed by adding a dynamic step size [5], and the accuracy of the solution is of 10− 9. Considering the nondifferentiation of absolute value equation, taking the advantages of the above algorithms and overcoming their disadvantages, a three-step iterative algorithm is designed for effectively solving AVEs (1) in this paper.
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