A New Filter Nonmonotone Adaptive Trust Region Method for Unconstrained Optimization

Xinyi Wang,Quan Qu,Xianfeng Ding

doi:10.3390/sym12020208

Abstract

In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is introduced. When a trial step is not successful, a multidimensional filter is employed to increase the possibility of the trial step being accepted. If the trial step is still not accepted by the filter set, it is possible to find a new iteration point along the trial step and the step length is computed by a fixed formula. The positive definite symmetric matrix of the approximate Hessian matrix is updated using the MBFGS method. The global convergence and superlinear convergence of the proposed algorithm is proven by some classical assumptions. The efficiency of the algorithm is tested by numerical results.

Highlights

Consider the following unconstrained optimization problem: minn f (x), x∈R (1)where f : Rn → R is a twice continuously differentiable function
At the iteration point xk, the trial step dk is obtained by the following quadratic subproblem: minmk (d) = gTk d + dT Bk d, n d∈R
The disadvantage of the traditional trust region method is that the subproblem needs to be solved many times to achieve an acceptable trial step in one iteration

Summary

Introduction

Where f : Rn → R is a twice continuously differentiable function. The trust region method is one of the prominent classes of iterative methods. The disadvantage of the traditional trust region method is that the subproblem needs to be solved many times to achieve an acceptable trial step in one iteration. To overcome this drawback, Mo et al [1] first proposed a nonmonotone trust region algorithm with a fixed step length. Compared with the standard nonmonotone algorithm, the new algorithm dynamically determines iterations based on filter elements and increases the possibility of the trial step being accepted This topic has received great attention in recent years (see [18,19,20,21]).

The New Algorithm

Convergence Analysis

Local Convergence

Preliminary Numerical Experiments

36 REVIEW

Conclusions