A Mixed Spectral CD‐DY Conjugate Gradient Method

Liu Jinkui,Du Xianglin,Wang Kairong

doi:10.1155/2012/569795

Liu Jinkui, Du Xianglin + Show 1 more

Open Access

https://doi.org/10.1155/2012/569795

Copy DOI

Abstract

A mixed spectral CD‐DY conjugate descent method for solving unconstrained optimization problems is proposed, which combines the advantages of the spectral conjugate gradient method, the CD method, and the DY method. Under the Wolfe line search, the proposed method can generate a descent direction in each iteration, and the global convergence property can be also guaranteed. Numerical results show that the new method is efficient and stationary compared to the CD (Fletcher 1987) method, the DY (Dai and Yuan 1999) method, and the SFR (Du and Chen 2008) method; so it can be widely used in scientific computation.

Highlights

The purpose of this paper is to study the global convergence properties and practical computational performance of a mixed spectral CD-DY conjugate gradient method for unconstrained optimization without restarts, and with appropriate conditions.Consider the following unconstrained optimization problem: min f x, 1.1 x∈Rn where f : Rn → R is continuously differentiable and its gradient g x ∇f x is available
We use the iterative method to solve 1.1, and the iterative formula is given by xk 1 xk αkdk, 1.2
Journal of Applied Mathematics where xk is the current iteration, αk is a positive scalar and called the step-size, which is determined by some line search, dk is the search direction defined by dk

Summary

Introduction

The purpose of this paper is to study the global convergence properties and practical computational performance of a mixed spectral CD-DY conjugate gradient method for unconstrained optimization without restarts, and with appropriate conditions. The conjugate direction method is a commonly used and effective method in optimization, and it only needs to use the information of the first derivative It overcomes the shortcoming of the steepest descent method in the slow convergence and avoids the defects of Newton method in storage and computing the second derivative. Based on the modified FR conjugate gradient method , Du and Chen proposed a new spectral conjugate gradient method: where dk They proved the global convergence of the modified spectral FR method In this paper, we call it SFR method with the mild conditions. Under some mild conditions, we give the global convergence of the mixed spectral CDDY conjugate gradient method with the Wolfe line search.

Algorithm and Lemmas

Global Convergence Property

Numerical Experiments