A scaled three-term conjugate gradient method for unconstrained optimization

Ibrahim Arzuka,Wah June Leong,Mohd R Abu Bakar

doi:10.1186/s13660-016-1239-1

Ibrahim Arzuka, Wah June Leong + Show 1 more

Open Access

https://doi.org/10.1186/s13660-016-1239-1

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Abstract

Conjugate gradient methods play an important role in many fields of application due to their simplicity, low memory requirements, and global convergence properties. In this paper, we propose an efficient three-term conjugate gradient method by utilizing the DFP update for the inverse Hessian approximation which satisfies both the sufficient descent and the conjugacy conditions. The basic philosophy is that the DFP update is restarted with a multiple of the identity matrix in every iteration. An acceleration scheme is incorporated in the proposed method to enhance the reduction in function value. Numerical results from an implementation of the proposed method on some standard unconstrained optimization problem show that the proposed method is promising and exhibits a superior numerical performance in comparison with other well-known conjugate gradient methods.

Highlights

In this paper, we are interested in solving nonlinear large scale unconstrained optimization problems of the form min f (x), x ∈ n, ( )where f : n → is an at least twice continuously differentiable function
Andrei [ ] considers the development of a three-term conjugate gradient method from the BFGS updating scheme of the inverse Hessian approximation restarted as an identity matrix at every iteration where the search direction is given by dk+
We proposed our three-term conjugate gradient method by incorporating the DFP updating scheme of the inverse Hessian approximation ( ), within the frame of a memoryless quasiNewton method where at each iteration the inverse Hessian approximation is restarted as a multiple of the identity matrix with a positive scaling parameter as sk sTk sTk yk μk yk yTk yTk yk and the search direction is given by dk+

Summary

Introduction

We are interested in solving nonlinear large scale unconstrained optimization problems of the form min f (x), x ∈ n, ( )where f : n → is an at least twice continuously differentiable function. An attractive property of these methods is that at each iteration, the search direction satisfies the descent condition, namely gkT dk = –c gk for some constant c > . Andrei [ ] considers the development of a three-term conjugate gradient method from the BFGS updating scheme of the inverse Hessian approximation restarted as an identity matrix at every iteration where the search direction is given by dk+

Results

Conclusion