Research On Numerical Algorithms For Solving The Absolute Value Equation

It is because the solution of AVE is strongly related to many important optimization problems and provides a new way to solve mathematical planning problems.Therefore,it is of great theoretical significance and practical application to study the effective solution methods for such problems.In this paper,different numerical algorithms for solving the AVE（1.1.1）are studied under the condition that the AVE has a unique solution based on the existing research results.In this paper,the asymptotic stability of the equilibrium point of the dynamical model,the gradient neural network algorithm,and the global convergence of the multi-variate spectral gradient algorithm is proved.Finally,this study validated the feasibility and effectiveness of these algorithms by means of numerical experiments.The main results of this paper are as follows:In Chapter 2,a new inverse-free dynamical model is developed for a class of AVEs（1.1.1）whose coefficient matrix A∈R⁹×9))is symmetric positive definite.It is also shown that the equilibrium point of this kinetic model is globally asymptotically stable.Based on the numerical experiments,the new inverse-free dynamical model is feasible.Meanwhile,by comparing with the five existing kinetic models,the analysis in terms of computational time and error shows that the inverse-free dynamical model proposed in this chapter is competitive.In Chapter 3,the approximate smooth functions of the absolute value operator|·|are discussed,focusing on the numerical performance of the convex combination of two smooth functions in the algorithm for solving AVE（1.1.1）.A gradient neural network model is established to solve the AVE（1.1.1）for the smoothed approximation model.And compare the smooth function after the convex combination with the existing smooth function by numerical experiments.In Chapter 4,the corresponding multivariate spectral gradient projection MSGP method is first proposed to solve for a special class of nonlinear equations,the AVE（1.1.1）.Compared with the existing modified multivariate spectral gradient method（MMSG）,the MSGP method uses a new search direction and an modified line search strategy,and introduces a projection step.Theoretical analysis shows that the MSGP method is globally convergent,and the feasibility and effectiveness of the MSGP method are verified by numerical experiments.