Some Researches On Algorithm For Absolute Value Equation

Absolute value equation(AVE), origining from interval linear equations and linear complementary problem, is equivalent to a nonsmooth optimization problem. In view of its simple and special structure, the study of absolute value equation causes the attention of many scholars. At present, research focuse on theory and algorithm. The former mainly studies the existence and uniqueness of the solution to the absolute value equation, while the latter mainly establishs efficient algorithm and make convergence analysis accordingly.For absolute value equation with unique solution, we present some methods for solving; while for absolute value equation with multiple solutions, we study the structure of solutions, and propose an algorithm for solving.The main contributions of this thesis can be summarized as follows:1. The existence of solution to absolute value equation was studied. Sufficient conditions of unique solution and 2n solutions are given. The relations between absolute value equation and linear complementary, and the relations between absolute value equation and boundary value problem of differential equation are analyzed.2. Feasible interior point algorithm and potential-reduction interior point algorithm are proposed for absolute value equation with unique solution. The convergence of these two algorithms is analyzed, and the preliminary numerical experiments are given.3. After upper and lower uniform approximation smooth function of absolute value function are established, absolute value equation is transformed into smooth optimization problem, and solved by smoothing Newton method. The convergence of method is shown, and the preliminary numerical experiments are given.4. An iterative method to absolute value equation is proposed, and the convergence analysis is given. Moreover, this method is applied to solve the second-order linearordinary differential equation with two-point boundary value problem. Numerical experiments show that this method has advantages of less number of iterations and high precision.5. The absolute value equation with 2n solutions is studied, and a method for computing all solutions is presented, then some issues for attention in solving absolute value equations with multiple solutions were pointed out.6. Firstly, an improved harmony search methods based on differential mutation operator is proposed, and population diversity is discussed in detail, then this method is applied to solve absolute value equation. Then, harmony search with worst and best strategy and global harmony search are proposed for solving absolute value equation. Finally, by using the thought of clustering and the characteristics of swarm intelligence algorithm, harmony search with clustering strategy is proposed solving absolute value equation with 2n solutions.