Finite algorithms for the numerical solutions of a class of nonlinear complementarity problems

Hongru Xu,Jinping Zeng

doi:10.1186/s13660-015-0781-6

Abstract

In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a nonlinear source term or some contact problems. Based on the reformulated systems of the piecewise almost linear equations, we propose a class of semi-iterative algorithms to find the exact solution of the problems. We prove that the semi-iterative algorithms enjoy a nice monotone convergence property in the sense that subsets of the indices consisting of the indices, for which the corresponding components of the iterates violate the constraints, become smaller and smaller. Then the algorithms converge monotonically to the exact solutions of the problems in a finite number of steps. Some numerical experiments are presented to show the effectiveness of the proposed algorithms.

Highlights

Let F : Rn → Rn be a given function
We denote the above problem by ALCP(A, φ) and call it an almost linear complementarity problem
ALCP(A, φ) has many applications, especially in engineering. It can be derived from the discrete simulations of Bratu obstacle problem [ ], which models the nonlinear diffusion phenomena taking place in combustion and in semiconductors, and of some free boundary problems with nonlinear source terms, which models the diffusion problems involving Michaelis-Menten or second order irreversible reactions [, ]

Summary

Introduction

Let F : Rn → Rn be a given function. The nonlinear complementarity problem, denoted by NCP(F, φ), is to find an x ∈ Rn such that x ≥ φ, F(x) ≥ , (x – φ)T F(x) = . ( . )In this paper, we focus on problem ( . ), in which the function F has the form of F(x) = Ax + (x), that is, we have the problem of finding an x ∈ Rn such that x ≥ φ, Ax + (x) ≥ , (x – φ)T Ax + (x) = , where A = (aij) ∈ Rn×n is a given matrix, φ = (φi) ∈ Rn is a given vector, and : Rn → Rn is a given diagonal differentiable mapping, that is, the ith component i of (x) is a function of the ith variable xi only:i = i(xi), i = , , . . . , n.Xu and Zeng Journal of Inequalities and Applications (2015) 2015:315We denote the above problem by ALCP(A, , φ) and call it an almost linear complementarity problem (see, e.g., [ ]). To solve the linear complementarity problem, the basic iteration of the active set strategies consists of two steps. In Section , semi-iterative algorithms are proposed for solving upper obstacle problems and mixed complementarity problems, respectively, via their PALS reformulations, and the monotone and finite convergence of the algorithms are obtained. Reduces to the semi-iterative Newton type algorithm for the solution of the linear complementarity problem

Results

Conclusion