Some generalized three-term conjugate gradient methods based on CD approach for unconstrained optimization problems

Ladan Arman,Long Liping,Yuanming Xu

doi:10.47577/technium.v3i2.1983

Ladan Arman, Long Liping + Show 1 more

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https://doi.org/10.47577/technium.v3i2.1983

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Abstract

Abstract In this paper, based on the efficient Conjugate Descent (CD) method, two generalized CD algorithms are proposed to solve the unconstrained optimization problems. These methods are three-term conjugate gradient methods which the generated directions by using the conjugate gradient parameters and independent of the line search satisfy in the sufficient descent condition. Furthermore, under the strong Wolfe line search, the global convergence of the proposed methods are proved. Also, the preliminary numerical results on the CUTEst collection are presented to show effectiveness of our methods.

Highlights

IntroductionWheref: Rn → R is a continuously differentiable function and its gradient g ≔ ∇f is available
Consider the following unconstrained optimization problem min f(x), x ∈ Rn (1)wheref: Rn → R is a continuously differentiable function and its gradient g ≔ ∇f is available.Conjugate Gradient (CG) methods are effective iterative methods for solving (1), especially for largescale problems
We introduce two three-term conjugate gradient methods based on Conjugate Descent (CD) algorithm

Summary

Introduction

Wheref: Rn → R is a continuously differentiable function and its gradient g ≔ ∇f is available. Conjugate Gradient (CG) methods are effective iterative methods for solving (1), especially for largescale problems. The important properties of these methods are the use only first-order derivatives, little storage and computation requirements, and strong local and global convergence properties [1, 9,18,22]. Where αk > 0 is step-length and usually obtained using some inexact line search. There are many variants of CG methods, which are obtained with different choices for the parameter βk. The most important CG methods proposed by Fletcher-. Reeves (FR) [16], Hestenes-Stiefel (HS) [19], Conjugate Descent (CD) by Fletcher [15], Polak-Ribiere-. Polyak (PRP) [22, 23], Dai-Yuan (DY) [10] and Hager-Zhang (HZ) [17] are defined by

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Motivation and the new algorithms

Convergence analysis

Numerical experiments

Findings

Conclusion