A Retrospective Filter Trust Region Algorithm for Unconstrained Optimization

Yue Lu,Zhongwen Chen

doi:10.4236/am.2010.13022

Abstract

In this paper, we propose a retrospective filter trust region algorithm for unconstrained optimization, which is based on the framework of the retrospective trust region method and associated with the technique of the multi-dimensional filter. The new algorithm gives a good estimation of trust region radius, relaxes the condition of accepting a trial step for the usual trust region methods. Under reasonable assumptions, we analyze the global convergence of the new method and report the preliminary results of numerical tests. We compare the results with those of the basic trust region algorithm, the filter trust region algorithm and the retrospective trust region algorithm, which shows the effectiveness of the new algorithm.

Highlights

Consider the following unconstrained optimization problem min f x (1)where x Rn, f : Rn R is a twice continuously differentiable function.The trust region method for unconstrained optimization is first presented by Powell [1], which, in some sense, is equivalent to the Levenberg-Marquardt method which is used to solve the least square problems and which was given by Levenberg [2] and Marquardt [3]
We propose a retrospective filter trust region algorithm for unconstrained optimization, which is based on the framework of the retrospective trust region method and associated with the technique of the multi-dimensional filter
The new algorithm gives a good estimation of trust region radius, relaxes the condition of accepting a trial step for the usual trust region methods

Summary

Introduction

Where x Rn , f : Rn R is a twice continuously differentiable function. The trust region method for unconstrained optimization is first presented by Powell [1], which, in some sense, is equivalent to the Levenberg-Marquardt method which is used to solve the least square problems and which was given by Levenberg [2] and Marquardt [3]. Gould et al [21] and Miao et al [22] applies the filter technique to unconstrained optimization, whose characteristic is to relax the condition of accepting a trial step for the usual trust region method, which improves the effectiveness of the algorithm in some sense. Bastin et al [25] presents a retrospective trust region method for unconstrained optimization Comparing their algorithm with the basic trust region algorithm, the updating way of the trust region radius is different, and the retrospective ratio k 1. In this paper we present a new algorithm for unconstrained optimization, which is based on the framework of the retrospective trust region method [25] and associated with the technique of the multi-dimensional filter [21,22].

Algorithm

Basic Assumptions and Lemmas

First Order Convergence

Second Order Convergence

Numerical Experiments