A Nonmonotone Adaptive Trust Region Method Based on Conic Model for Unconstrained Optimization

Zhaocheng Cui

doi:10.1155/2014/237279

Abstract

We propose a nonmonotone adaptive trust region method for unconstrained optimization problems which combines a conic model and a new update rule for adjusting the trust region radius. Unlike the traditional adaptive trust region methods, the subproblem of the new method is the conic minimization subproblem. Moreover, at each iteration, we use the last and the current iterative information to define a suitable initial trust region radius. The global and superlinear convergence properties of the proposed method are established under reasonable conditions. Numerical results show that the new method is efficient and attractive for unconstrained optimization problems.

Highlights

In this paper, we consider the following unconstrained optimization problem: min f (x), x∈Rn (1)where f: Rn → R is a continuously differentiable function
We propose a nonmonotone adaptive trust region method for unconstrained optimization problems which combines a conic model and a new update rule for adjusting the trust region radius
We propose a new adaptive trust region method which combines the conic model and nonmonotone technique which was given in [33] and a new update rule for adjusting the trust region radius

Summary

Introduction

We consider the following unconstrained optimization problem: min f (x) , x∈Rn (1). Yuan [2], Nocedal, and Yuan [3] proposed various trust region methods for optimization problems It calculates a trial step by solving the subproblem min φk (d). The numerical results show that the adaptive trust region algorithms [21,22,23,24] are more effective than traditional trust region methods for unconstrained optimization problems. We propose a new adaptive trust region method which combines the conic model and nonmonotone technique which was given in [33] and a new update rule for adjusting the trust region radius. We describe a new nonmonotone adaptive trust region method based on conic model for unconstrained optimization problems.

Algorithm Description

Convergence Analysis

Superlinear Convergence

Numerical Results

Conclusions