Research On Numerical Algorithm Of Absolute Value Equation

With the rapid development of modern science and technology and the advent of the era of big data,optimization theory is widely used to solve various problems in industry,agriculture,finance,management,communication and other fields,especially the non-differentiable optimization problem,which has more important theoretical research significance and practical application value in Engineering application and scientific calculation,It becomes an important branch of optimization theory.The absolute value equation studied in this paper is a special non-differentiable optimization problem.Most of the research on absolute value equation focuses on the theory and algorithm of the solution.The former mainly focuses on the existence and uniqueness of the solution of absolute value equation,while the latter mainly constructs an effective algorithm to solve the absolute value equation.This paper focuses on the solution of absolute value equation.Therefore,we first briefly introduce some theoretical researches on the solution of absolute value equation.Secondly,some classical algorithms for solving absolute value equation are introduced.These algorithms can be summarized into two branches,which are inspired by the ideas of solving linear problems and nonlinear problems,and then put forward the corresponding algorithms for solving the absolute value equations.The methods to solve linear problems include residual iterative method,relaxed nonlinear PHSS-like iteration method,SOR-like iteration method and so on.The methods to solve nonlinear problems include generalized Newton method,smooth Newton method,Levenberg-Marquardt method and so on.This paper first proposes a modified SOR-like method for solving absolute value equations,and analyzes the theory of the modified SOR-like method,gives sufficient conditions to ensure the convergence of the method,and also discusses the optimal value expression of parameters in the new method.Compared with SOR-like method,the optimal parameters can improve the CPU execution time consumption,iteration steps,residual error and solution error of the modified method.At the same time,the conditions for the modified SOR-like method to obtain the optimal parameters are weaker than those for the SOR-like method.Secondly,in order to solve the absolute value equation,this paper proposes a preconditioned AOR iterative method,studies the convergence of the preconditioned AOR iterative method,and gives the corresponding comparison theorem.The comparison theorem points out that when the original AOR iterative method converges or diverges,the convergence and divergence of the preconditioned AOR iterative method are the same;In addition,the influence of two different preconditioners on the efficiency of AOR iterative method is explored.Finally,the numerical simulation results show that the conclusion of the comparison theorem is reasonable and the preconditioned AOR iterative method is feasible and effective.