Some Algorithms For Solving A Class Of Absolute Value Equations

The absolute value equations is a kind of NP-hard and non-differentiable problem,because it is equivalent to many other mathematical problems(such as generalized linear complementarity problem,standard linear complementarity problem,bilinear programming problem,concave minimization problem,etc).And,compared with these problems,the absolute value equations have the advantages of simple structure,good transformation,etc.And it has attracted the attention of scholars.In this paper,for a class of absolute value equations with the form of Ax-|x| = b when the solution exists,some fast and efficient solution algorithms are designed from different approaches,and its convergence is proved.The main research results are summarized as follows:1.Based on the generalized Newton algorithm,we have effectively improved and mainly add a dynamic step size to the search direction.The improved algorithm has global linear convergence,numerical experiments show that the improved algorithm is much faster and more accurate than the generalized Newton algorithm,with less number of iterations required.2.We are inspired by some iterative algorithms for solving one-dimensional nonlinear equations.Let's first convert the absolute value equation to a nonlinear equation,the methods are extended to the n-dimension from the one-dimension,we designed two-step and threestep iterative algorithms,these two methods have global linear convergence.The numerical experiments show that the improved algorithms have fast convergence speed,high solution precision and few iterations.3.In order to overcome the shortcoming of particle swarm optimization in the later period,when the diversity of the seeker becomes worse,the convergence speed becomes slower,the calculation is stagnant,it is easy to fall into the local optimum and cannot jump out,and the following improvements are made.Our first improvement strategy is to design the inertia weight from the previous constant change to decrease the exponential trend with the increase of the number of iterations.The second improvement is to embed the pattern search algorithm with strong local mining ability into the particle swarm algorithm.In the early iteration,the particle swarm optimization algorithm is mainly used for global search,and the strong local convergence performance of the pattern search algorithm is played later.The two algorithms are used interchangeably,taking advantages and disadvantages respectively.The third improvement is to set the self-learning factor to a linear decreasing trend,and the social learning factor is designed to be a linearly increasing trend.Through the standard test functions and absolute value equations to verify the above three improved strategies,the calculation level of the three algorithms has been greatly improved,especially in the later stage,the particle's local optimization ability has been increased,and the particle's ability to jump out of the local best trap has been stimulated.In general,the local mining ability and global searching ability of the algorithm have been well balanced.4.In order to improve the seeker optimization algorithm in the late stage of search precocity,easy to fall into the local optimal can not jump out shortcomings.We introduce the simplex search and the pattern search respectively.After running seeker optimization algorithm for a certain number of algebras,we start to implement the two algorithms separately to increase the local part of the algorithm.To increase the local mining level in the later stage of the algorithm and jump out of the trap of local optimization,the experiment proves that the improved two algorithms indeed increase the local search level of the seeker optimization algorithm,thus improving the convergence speed,solution accuracy of the whole algorithm.