A smoothing and regularization predictor-corrector method for nonlinear inequalities

Haitao Che

doi:10.1186/1029-242x-2012-214

Abstract

For a system of nonlinear inequalities, we approximate it by a family of parameterized smooth equations via a new smoothing function. We present a new smoothing and regularization predictor-corrector algorithm. The global and local superlinear convergence of the algorithm is established. In addition, the smoothing parameter μ and the regularization parameter e in our algorithm are viewed as different independent variables. Preliminary numerical results show the efficiency of the algorithm. MSC: 90C33; 90C30; 15A06

Highlights

In Section , we present a smoothing and regularization predictor-corrector method for solving the nonlinear inequalities and establish the global and local convergence of the proposed algorithm
Consider the following system of nonlinear inequalities: f (x) ≤, ( . )where f (x) = (f (x), f (x), . . . , fn(x)) and fi : Rn → R is a continuously differentiable function for i =, . . . , n
Among various solution methods for the inequality problems [ – ], the smoothing-type methods receive much attention [ – ] which first transform the problem as a system of nonsmooth equations and approximate it by a smooth equation and solve it by the smoothing Newton methods

Summary

Introduction

In Section , we present a smoothing and regularization predictor-corrector method for solving the nonlinear inequalities and establish the global and local convergence of the proposed algorithm. [ ] Suppose that φ : Rn → Rm is a locally Lipschitz function semi-smooth at x. This new smoothing function has the following properties. (b) If f is a P -function, H (z) is nonsingular at any R++ × R++ × Rn. Proof (a) is straightforward, so we only prove (b).

Results

Conclusion