A New Noninterior Continuation Method for Solving a System of Equalities and Inequalities

Jianguang Zhu,Binbin Hao

doi:10.1155/2014/592540

Abstract

By using slack variables and minimum function, we first reformulate the system of equalities and inequalities as a system of nonsmooth equations, and, using smoothing technique, we construct the smooth operator. A new noninterior continuation method is proposed to solve the system of smooth equations. It shows that any accumulation point of the iteration sequence generated by our algorithm is a solution of the system of equalities and inequalities. Some numerical experiments show the feasibility and efficiency of the algorithm.

Highlights

In this paper, we consider the following system of equalities and inequalities: fI (x) ≤ 0, (1)fE (x) = 0, where I = {1, . . . , m} and E = {m + 1, . . . , n}
We show that the proposed algorithm is globally linearly convergent
We report some preliminary numerical results, which demonstrate that the algorithm is effective for solving (1)

Summary

Introduction

We consider the following system of equalities and inequalities: fI (x) ≤ 0, (1). In. Section 3, we propose a noninterior continuation method for solving (1). For a continuously differentiable function F : Rn → Rm, we denote the Jacobian of F at x ∈ Rn by F󸀠(x). One well-known NCP function is the minimum function [9], which is defined as follows: φmin (a, b) = a + b − |a − b|. Based on the minimum function, we reformulate (5) into the following equivalent system of nonlinear equation: fE (x). Since the function in (6) is nonsmooth, the noninterior continuation method cannot be directly applied to solve (6). In order to make (6) solvable by the noninterior continuation method, we will use the smoothing technique and construct the smooth approximation of Φ as Φμ.

Algorithm

Convergence of Algorithm 1

Numerical Experiments