A Study Of The Numerical Solution For The Generalized Absolute Value Equation

The absolute value equation is an NP-hard optimization problem.It is derived from two classical problems: the interval linear equation and the linear complementarity problem.The interval linear equation means that the elements in its coefficient matrix are not constant,but are located in a certain interval range;the linear complementary problem is derived from a class of optimization problems in computational mechanics,and includes classical linear programming and quadratic programming.Linear complementary problems and absolute value problems can be transformed into each other,and because the structure of absolute value problems is more concise,solving absolute value equations becomes a lot of problems.The hot spot that scholars pay attention to.This article starts from the source of the absolute value equation and is mainly divided into the following parts:The first chapter introduces the background of absolute value equation: linear complementary problem,mainly introduces the source of linear complementary problem and several optimization problems related to it,and a specific mechanical problem: reinforced concrete problem is given to derive the linear complementary problem.Finally,introduces the theoretical research status of the absolute value equation,mainly including the sufficient condition(easy to verify)and the necessary and sufficient condition of the uniqueness of the solution,and explained the idea of sufficient conditions from a visual point of view.Chapter 2 introduces the Douglas-Rachford splitting algorithm for solving generalized absolute value equations,its core is to construct a fixed point iteration.Since the system of linear equations needs to be solved during the iteration,we also show that the iteration converges within a certain tolerance,therefore,an inexact version of the algorithm is proposed,which reduces the complexity of the solution.Chapter 3 presents a modified version of the Barzilai-Borwein algorithm for solving generalized absolute value equations.The essence of the algorithm is a variant of the gradient descent method,the biggest advantage is that it proposes a method for automatically solving the iterative step size,and in the quadratic optimization problem only two steps are required to terminate iteratively.Based on the Barzilai-Borwein algorithm,we propose a modified version of it and prove that convergence.Chapter 4 introduces the algorithm for solving generalized absolute value equations based on dynamical systems.The core turns the problem of solving equations into solving dynamical system equilibrium point,and then prove the stability of the constructed system according to the Lyapunov stability theory,and finally by the numerical experiments we proved the effectiveness of the algorithm.Chapter 5 is a general summary of the full text,and further puts forward the prospect of improving the performance of the algorithm.