Abstract

The spectral conjugate gradient methods are very interesting and have been proved to be effective for strictly convex quadratic minimisation. In this paper, a new spectral conjugate gradient method is proposed to solve large-scale unconstrained optimisation problems. Motivated by the advantages of approximate optimal stepsize strategy used in the gradient method, we design a new scheme for the choices of the spectral and conjugate parameters. Furthermore, the new search direction satisfies the spectral property and sufficient descent condition. Under some suitable assumptions, the global convergence of the developed method is established. Numerical comparisons show better behaviour of the proposed method with respect to some existing methods for a set of 130 test problems.

Highlights

  • Consider the following unconstrained optimisation: min f (x), x ∈ Rn, (1)where f : Rn → R is continuously differentiable and bounded from below

  • Due to the strong convergence of the Newton method, Andrei [1] proposed an accelerated conjugate gradient method, which took advantage of the Newton method to improve the performance of the conjugate gradient method

  • We propose a new spectral conjugate gradient method based on the idea of the approximate optimal stepsize

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Summary

Introduction

Where f : Rn → R is continuously differentiable and bounded from below. Conjugate gradient method is one of the most effective line search methods for solving unconstrained optimisation problem (1) due to its features of low memory requirement and simple computation. Based on the descent condition, Wan et al [29] and Zhang et al [35] presented the modified PRP and FR spectral conjugate gradient method, respectively. Liu et al [18, 19] introduced approximate optimal stepsizes (αkAOS) for gradient method. By minimising φ(α), they obtained αkAOS gk 2 gkTBk gk and proposed the approximate optimal gradient methods. We propose a new spectral conjugate gradient method based on the idea of the approximate optimal stepsize. Compared with the SCG method [5], the proposed method generates the sufficient descent direction per iteration and does not require more computation costs.

The new spectral conjugate gradient algorithm
Findings
Conclusions

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