Global Convergence of a New Nonmonotone Filter Method for Equality Constrained Optimization

Ke Su,Wei Liu,Xiaoli Lu

doi:10.1155/2013/976509

Abstract

A new nonmonotone filter trust region method is introduced for solving optimization problems with equality constraints. This method directly uses the dominated area of the filter as an acceptability criterion for trial points and allows the dominated area decreasing nonmonotonically. Compared with the filter-type method, our method has more flexible criteria and can avoid Maratos effect in a certain degree. Under reasonable assumptions, we prove that the given algorithm is globally convergent to a first order stationary point for all possible choices of the starting point. Numerical tests are presented to show the effectiveness of the proposed algorithm.

Highlights

We analyze an algorithm for solving optimization problems where a smooth objective function is to be minimized subject to smooth nonlinear equality constraints
There are many trust region methods for equality constrained nonlinear programming (P), for example, Byrd et al [1], Dennis Jr. et al [2] and Powell and Yuan [3], but in these works, a penalty or augmented Lagrange function is always used to test the acceptability of the iterates
Filter technique has been employed to many approaches, for instance, SLP methods [5], SQP methods [6,7,8], interior point approaches [9], bundle techniques [10], and so on

Summary

Introduction

We analyze an algorithm for solving optimization problems where a smooth objective function is to be minimized subject to smooth nonlinear equality constraints. There are many trust region methods for equality constrained nonlinear programming (P), for example, Byrd et al [1], Dennis Jr. et al [2] and Powell and Yuan [3], but in these works, a penalty or augmented Lagrange function is always used to test the acceptability of the iterates. Ulbrich [12] proposed a class of penalty-function-free nonmonotone trust region methods for nonlinear equality constrained optimization without filter technique. Motivated by the ideas and methods above, we propose a modified nonmonotone filter trust region method for solving problem (P).

The Fraction of Cauchy Decrease and the Composite SQP Step

A New Nonmonotone Filter Technique

The New Nonmonotone Filter Trust Region Algorithm

The Convergence Properties

Some Numerical Experiments