An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization

Liu, Zexian; Liu, Hongwei

doi:10.1007/s11075-017-0365-2

Cited by 30 publications

(18 citation statements)

References 24 publications

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“…Liu et al [19] proposed GM_AOS 1, GM_AOS 2 and GM_AOS 3 algorithms, and GM_AOS 2 algorithm was slightly better than the other algorithms. When the quadratic model is considered, the algorithm developed by [18] is identical with GM_AOS 1 algorithm. In a certain sense, our algorithm can be viewed as an extension of SCG algorithm [5] and a modification of DY algorithm [8].…”

Section: Numerical Resultsmentioning

confidence: 99%

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The new spectral conjugate gradient method for large-scale unconstrained optimisation

et al. 2020

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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.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Recently, Liu et al [18,19] introduced approximate optimal stepsizes (α AOS k ) for gradient method. They constructed a quadratic approximation model of…”

Section: Introductionmentioning

confidence: 99%

The new spectral conjugate gradient method for large-scale unconstrained optimisation

et al. 2020

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“…In [53], a new concept of the approximate optimal step size for gradient method is introduced and used to interpret the BB method; an efficient gradient method with the approximate optimal step size for unconstrained optimization is presented. The next definition is introduced in [ The approximate optimal step size is different from the steepest descent step size, which will lead to the expensive computational cost.…”

Section: Algorithm 125 (Barzilai-borwein Gradient Method Ie Bbmentioning

confidence: 99%

Some Unconstrained Optimization Methods

Djordjević¹

2019

Applied Mathematics

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Although it is a very old theme, unconstrained optimization is an area which is always actual for many scientists. Today, the results of unconstrained optimization are applied in different branches of science, as well as generally in practice. Here, we present the line search techniques. Further, in this chapter we consider some unconstrained optimization methods. We try to present these methods but also to present some contemporary results in this area.

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“…We see from the definition of approximately optimal stepsize that the numerical performance of gradient method with approximately optimal stepsize depends heavily on approximation model φ k (α). Some gradient methods with approximately optimal stepsizes [31][32][33][34] were later proposed for unconstrained optimization, and the numerical results in [31][32][33][34] suggested that these gradient methods with approximately optimal stepsizes are surprisingly efficient.…”

Section: Introductionmentioning

confidence: 99%

“…In those gradient methods with approximately optimal stepsizes [31][32][33][34], the gradient method with approximately optimal stepsizes based on conic model [31] has enjoyed some attentions [35] due to its good nice numerical performance. In this paper, we present an improved gradient method with approximately optimal optimal optimal stepsize based on conic model for unconstrained optimization.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Gradient Method with Approximately Optimal Stepsize Based on Tensor Model for Unconstrained Optimization

Liu

2019

J Optim Theory Appl

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A new type of stepsize, which was recently introduced by Liu and Liu (Optimization, 67(3), [427][428][429][430][431][432][433][434][435][436][437][438][439][440] 2018), is called approximately optimal stepsize and is quit efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, based on the work (Numer. Algorithms 78(1), 21-39, 2018), we present an improved gradient method with approximately optimal stepsize based on conic model for unconstrained optimization. If the objective function f is not close to a quadratic on the line segment between the current and latest iterates, we construct a conic model to generate approximately optimal stepsize for gradient method if the conic model can be used; otherwise, we construct some quadratic models to generate approximately optimal stepsizes for gradient method. The convergence of the proposed method is analyzed under suitable conditions. Numerical comparisons with some well-known conjugate gradient software packages such as CG DESCENT (SIAM J. Optim. 16(1), 170-192, 2005) and CGOPT (SIAM J. Optim. 23(1), 296-320, 2013) indicate the proposed method is very promising.Keywords Approximately optimal stepsize · Barzilai-Borwein method · Quadratic model · Conic model · BFGS update formula · Gradient method with approximately optimal stepsize Mathematics Subject Classification (2000) 90C06 · 65K

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An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization

Cited by 30 publications

References 24 publications

The new spectral conjugate gradient method for large-scale unconstrained optimisation

The new spectral conjugate gradient method for large-scale unconstrained optimisation

Some Unconstrained Optimization Methods

An Efficient Gradient Method with Approximately Optimal Stepsize Based on Tensor Model for Unconstrained Optimization

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