Algorithm 813

Birgin, Ernesto G.; Martínez, José Mário; Raydan, Marcos

doi:10.1145/502800.502803

Cited by 222 publications

(146 citation statements)

References 27 publications

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“…In this subsection, we compare the performance of our ACBB stepsize algorithm, denoted ACBB, with the SPG2 algorithm of Birgin et al (2000Birgin et al ( , 2001, with the PRP+ conjugate gradient code developed by Gilbert & Nocedal (1992) and with the CG DESCENT code of Hager & Zhang (2005b, to appear). The SPG2 algorithm is an extension of Raydan's (1997) GBB algorithm which was downloaded from the TANGO web page maintained by Ernesto Birgin.…”

Section: Numerical Resultsmentioning

confidence: 99%

The cyclic Barzilai-–Borwein method for unconstrained optimization

Dai¹,

Hager²,

Schittkowski³

et al. 2006

IMA Journal of Numerical Analysis

174

163

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In the cyclic Barzilai-Borwein (CBB) method, the same Barzilai-Borwein (BB) stepsize is reused for m consecutive iterations. It is proved that CBB is locally linearly convergent at a local minimizer with positive definite Hessian. Numerical evidence indicates that when m > n/2 3, where n is the problem dimension, CBB is locally superlinearly convergent. In the special case m = 3 and n = 2, it is proved that the convergence rate is no better than linear, in general. An implementation of the CBB method, called adaptive cyclic Barzilai-Borwein (ACBB), combines a non-monotone line search and an adaptive choice for the cycle length m. In numerical experiments using the CUTEr test problem library, ACBB performs better than the existing BB gradient algorithm, while it is competitive with the well-known PRP+ conjugate gradient algorithm.

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Section: Numerical Resultsmentioning

confidence: 99%

The cyclic Barzilai-–Borwein method for unconstrained optimization

Dai¹,

Hager²,

Schittkowski³

et al. 2006

IMA Journal of Numerical Analysis

174

163

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“…Throughout this paper we assume that f is defined and has continuous partial derivatives on an open set that contains Ω. The Spectral Projected Gradient (SPG) method [6,7] was recently proposed for solving (1), especially for large-scale problems since the storage requirements are minimal. This method has proved to be effective for very large-scale convex programming problems.…”

Section: Introductionmentioning

confidence: 99%

Inexact spectral projected gradient methods on convex sets

Birgin¹

2003

IMA Journal of Numerical Analysis

166

178

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A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

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“…without any additional assumption on boundedness of level sets). Results of this kind for the convex case can be found in [8], [10] and [14] for the method with exogenously given β k 's satisfying (4)-(5), in [12] for the method with exact lineasearches as in (6), and in [2], [7] and [13] for the method with the Armijo rule (7)- (8). We observe that in the case of β k 's given by (4)-(5) the method is not in general a descent one, i.e.…”

Section: The Steepest Descent Methodsmentioning

confidence: 99%

“…[5], [6], [4]. In this method β k is taken as a safeguarded spectral parameter, with the following meaning.…”

mentioning

confidence: 99%

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On the convergence properties of the projected gradient method for convex optimization

Iusem¹

2003

Mat. apl. comput.

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Abstract. When applied to an unconstrained minimization problem with a convex objective, the steepest descent method has stronger convergence properties than in the noncovex case: the whole sequence converges to an optimal solution under the only hypothesis of existence of minimizers (i.e. without assuming e.g. boundedness of the level sets). In this paper we look at the projected gradient method for constrained convex minimization. Convergence of the whole sequence to a minimizer assuming only existence of solutions has also been already established for the variant in which the stepsizes are exogenously given and square summable. In this paper, we prove the result for the more standard (and also more efficient) variant, namely the one in which the stepsizes are determined through an Armijo search. 90C25, 90C30. Mathematical subject classification:

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Algorithm 813

Cited by 222 publications

References 27 publications

The cyclic Barzilai-–Borwein method for unconstrained optimization

The cyclic Barzilai-–Borwein method for unconstrained optimization

Inexact spectral projected gradient methods on convex sets

On the convergence properties of the projected gradient method for convex optimization

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