Efficient hybrid conjugate gradient techniques

Storey, C.; Touati-Ahmed, D.

doi:10.1007/bf00939455

Cited by 215 publications

(85 citation statements)

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“…Many authors have presented other choices for the scalar β k , for example Buckley and Lenir [2], Daniel [13], Gilbert and Nocedal [16], Qi et al [28], Shanno [29], and Touati-Ahmed and Storey [30]. Observing that the formulae (1.6)-(1.9), (1.12) and (2.1) share two numerators and three denominators, we can use combinations of these numerators and denominators to obtain the following three-parameter family:…”

Section: A Three-parameter Family Of Conjugate Gradient Methodsmentioning

confidence: 99%

A three-parameter family of nonlinear conjugate gradient methods

Dai¹,

Yuan²

2000

Math. Comp.

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Abstract. In this paper, we propose a three-parameter family of conjugate gradient methods for unconstrained optimization. The three-parameter family of methods not only includes the already existing six practical nonlinear conjugate gradient methods, but subsumes some other families of nonlinear conjugate gradient methods as its subfamilies. With Powell's restart criterion, the three-parameter family of methods with the strong Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the three-parameter family of methods. This paper can also be regarded as a brief review on nonlinear conjugate gradient methods.

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Section: A Three-parameter Family Of Conjugate Gradient Methodsmentioning

confidence: 99%

A three-parameter family of nonlinear conjugate gradient methods

Dai¹,

Yuan²

2000

Math. Comp.

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“…Hybrid conjugate gradient algorithms using projections: hybrid Dai-Yuan [23], Gilbert and Nocedal [28], Hu and Storey [34], Touati-Ahmed and Storey [50], hybrid Liu and Storey [36], and hybrid conjugate gradient algorithms using the concept of convex combination of classical schemes: convex combination of Hestenes-Stiefel and Dai-Yuan with Newton direction [3,4,8], convex combination of Polak-Ribière-Polyak and DaiYuan with conjugacy condition [7]. Scaled BFGS preconditioned conjugate gradient algorithms by Shanno [47,48], Birgin and Martínez [18] and Andrei [2,9].…”

Section: Classical Conjugate Gradient Algorithmsmentioning

confidence: 99%

A Numerical Study on Efficiency and Robustness of Some Conjugate Gradient Algorithms for Large-scale Unconstrained Optimization

Andrei¹

2013

SIC

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Abstract:A numerical evaluation and comparisons using performance profiles of some representative conjugate gradient algorithms for solving a large variety of large-scale unconstrained optimization problems are carried on. In this intensive numerical study we selected eight known conjugate gradient algorithms: Hestenes and Stiefel (HS), Polak-Ribière-Polyak (PRP), CONMIN, ASCALCG, CG-DESCENT, AHYBRIDM, THREECG and DESCON. These algorithms are different in many respects. However, they have a lot of concepts in common, which give the numerical comparisons sense and confident expectations. The initial search direction in all algorithms is the negative gradient computed in the initial point and the step length is computed by the Wolfe line search conditions. Excepting CONMIN and CG-DESCENT, all the algorithms from this numerical study implement an acceleration scheme which modifies the step length in a multiplicative manner to improve the reduction of the functions values along the iterations. The numerical study is based on a set of 800 artificially large-scale unconstrained optimization test functions of different complexity and with different structures of their Hessian matrix. A detailed numerical evaluation based on performance profiles is applied to the comparisons of these algorithms showing that all of them are able to solve difficult large-scale unconstrained optimization problems. However, comparisons using only artificially test problems are weak and dependent on arbitrary choices concerning the stopping criteria of the algorithms and on decision of whether an algorithm found a solution or not. To get definitive conclusions using this sort of comparisons based only on artificially test problems is an illusion. However, using some real unconstrained optimization applications we can get a more confident conclusion about the efficiency and robustness of optimization algorithms considered in this numerical study.

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“…Then In the above algorithm, a back-tracking line search method is used and the step length α k to be the first element of the sequence: 1, , ..., 2 −i , ... that satisfies a sufficient decrease condition, see, e.g., Nash and Sofer [48] for details. This hybrid nonlinear conjugate gradient method was first proposed by Touati-Ahmed and Storey [61]. It has been tested to perform well for many difficult numerical examples [24] in comparison with the Polak-Ribiére-Polyak method and it does not require the line search scheme to satisfy the strong Wolfe condition.…”

Section: Numerical Solutionsmentioning

confidence: 99%