Abstract
In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.
Highlights
Consider the following absolute value problem [1]-[3]: Ax − x = 0 (1)where A∈ Rn×n, x,b ∈ R, x is absolute value of x, it is a subclass of absolute value equations Ax − B x = b which is proposed by Rohn [4], and it is a NP-hard problem [1].The AVE has closed relation with some important problems, for example, the linear programming, Quadratic programming problem and the bimatrix game problem
We give some smooth of numerical tests of neural network algorithm, due to the complementarity problem can be transformed to absolute value equations, we consider the linear complementarity problem which is equivalent to the absolute value equations as test cases
In order to obtain the approximate solution of the original problem we use the proposed neural network model to solve the unconstrained optimization problem
Summary
The above problems can be transformed into the linear complementarity problem, and the linear complementarity problem can be transformed into the absolute value equations. Due to its simple and special structure and application value, the research on absolute value equation has drawn attention of many researchers. (2015) A Smoothing Neural Network Algorithm for Absolute Value Equations. We present a smooth approximation function which is based on neural network method to solve the AVE. By using a smooth approximation function of x , we turn it into a differentiable unconstrained optimization problem. We obtain the approximate solution of the original problem based on our established unconstrained optimization problem and the neural network model. Compared with the Newton method, the neural network model needs less requirement for the hardware of compute and the iterative process is real-time
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