An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints

Cong Zhang,Zhibin Zhu,Limin Sun,Minglei Fang

doi:10.4236/am.2015.610152

Cong Zhang, Zhibin Zhu + Show 2 more

Open Access

https://doi.org/10.4236/am.2015.610152

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Journal: Applied mathematics	Publication Date: Jan 1, 2015
Citations: 3	License type: CC BY 4.0

Affiliation: Xinyang Normal University, Guilin University

Abstract

This paper discusses a special class of mathematical programs with equilibrium constraints. At first, by using a generalized complementarity function, the discussed problem is transformed into a family of general nonlinear optimization problems containing additional variable μ. Furthermore, combining the idea of penalty function, an auxiliary problem with inequality constraints is presented. And then, by providing explicit searching direction, we establish a new conjugate projection gradient method for optimization with nonlinear complementarity constraints. Under some suitable conditions, the proposed method is proved to possess global and superlinear convergence rate.

Highlights

Mathematical programs with equilibrium constraints (MPEC) include the bilevel programming problem as its special case and have extensive applications in practical areas such as traffic control, engineering design, and economic modeling
We consider an important subclass of MPEC problem, which is called mathematical program with nonlinear complementarity constraints (MPCC): min f ( x, y) s.t. g ( x, y) ≤ 0, (1.1)
The global convergence is obtained as well as the superlinear convergence rate

Summary

Introduction

Mathematical programs with equilibrium constraints (MPEC) include the bilevel programming problem as its special case and have extensive applications in practical areas such as traffic control, engineering design, and economic modeling. We consider an important subclass of MPEC problem, which is called mathematical program with nonlinear complementarity constraints (MPCC): min f ( x, y) s.t. g ( x, y) ≤ 0,. The following practical results about function φ hold:. Similar to [12], we define the following penalty function ( ) ( ) ( ) m θc ( x, y, w, μ=) f ( x, y) − c∑ φ y j , wj , μ + c j ( x, y, w) + c eμ −1 , j=1 where c > 0 is a penalty parameter.

Preliminaries and Algorithm

Global Convergence

Conclusion