On the Identification Property of a Projected Gradient Method

Angelis, P. L. De; Toraldo, Gerardo

doi:10.1137/0730077

Cited by 9 publications

(5 citation statements)

References 18 publications

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“…The studies in [7] and [13] show that this class of methods has stronger ability to indentify the optimal face than those involving the test (4.1). In this paper we will consider the use of (4.1) since our next object is to design efficient projected gradient methods for quadratic programming problems subject to box constraints and one single linear constraint where the projection onto the feasible set is not so cheap.…”

Section: An Adaptive Nonmonotone Line Searchmentioning

confidence: 99%

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

2005

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Summary. This paper studies projected Barzilai-Borwein (PBB) methods for large-scale box-constrained quadratic programming. Recent work on this method has modified the PBB method by incorporating the Grippo-Lampariello-Lucidi (GLL) nonmonotone line search, so as to enable global convergence to be proved. We show by many numerical experiments that the performance of the PBB method deteriorates if the GLL line search is used. We have therefore considered the question of whether the unmodified method is globally convergent, which we show not to be the case, by exhibiting a counter example in which the method cycles. A new projected gradient method (PABB) is then considered that alternately uses the two BarzilaiBorwein steplengths. We also give an example in which this method may cycle, although its practical performance is seen to be superior to the PBB method. With the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods, we examine other recent work on nonmonotone line searches, and propose a new adaptive variant with some attractive features. Further numerical experiments show that the PABB method with the adaptive line search is the best BB-like method in the positive definite case, and it compares reasonably well against the GPCG algorithm of Moré and Toraldo.

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Section: An Adaptive Nonmonotone Line Searchmentioning

confidence: 99%

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

2005

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“…The following proposition shows that the iterates of any gradient method can be written in the form (15). We note that expression (17) given in the proposition has been reported also in [10].…”

Section: Filter Factor Analysismentioning

confidence: 82%

“…Therefore, there has been an increasing interest in the development of projected methods able to effectively solve such constrained problems (see, e.g., [8,19,32,4,5,31]). For this reason, we intend to investigate the behavior of SDA, SDC, and other efficient gradient methods with regularization properties, within projected gradient frameworks such as those discussed in [33,15].…”

Section: Discussionmentioning

confidence: 99%

On the regularizing behavior of the SDA and SDC gradient methods in the solution of linear ill-posed problems

Asmundis

Serafino

Landi

2016

Journal of Computational and Applied Mathematics

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“…For completeness, we also run the experiments on the strictly convex problems with nondegenerate solutions by replacing the line search strategy in PABB min with a monotone line search along the feasible direction [2, Section 2.3.1], which requires only one projection per GP iteration. We note that this line search does not guarantee 25 in general that the sequence generated by the GP method identifies in a finite number of steps the variables that are active at the solution (see, e.g., [13]). Nevertheless, we made experiments with the line search along the feasible direction, to see if it may lead to any time gain in practice.…”

Section: 3mentioning

confidence: 99%

A Two-Phase Gradient Method for Quadratic Programming Problems with a Single Linear Constraint and Bounds on the Variables

Serafino¹,

Toraldo²,

Viola³

et al. 2018

SIAM J. Optim.

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We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for boundconstrained convex quadratic programming [J.J. Moré and G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportional iterate, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.Gradient Projection (GP) methods are widely used to solve large-scale SLBQP problems, thanks to the availability of low-cost projection algorithms onto the feasible set of (1.1) (see, e.g, [8,12,9]). In particular, Spectral Projected Gradient methods [4], other GP methods exploiting variants of Barzilai-Borwein (BB) steps [35,12], and more recent Scaled Gradient Projection methods [6] have proved their effectiveness in several applications.Bound-constrained Quadratic Programming problems (BQPs) can be regarded as a special case of SLBQPs, where the theory or the implementation can be simplified. This has favoured the development of more specialized gradient-based methods, built upon the idea of combining steps aimed at identifying the variables that are active at a solution (or at a stationary point) with unconstrained minimizations in reduced spaces defined by fixing the variables that are estimated active [31,24,32,25,3,20,22,27,21,28]. Thanks to the identification properties of the GP method [7] and to its capability of adding/removing multiple variables to/from the active set in a single iteration, GP steps are a natural choice to determine the active variables. A wellknown method based on this approach is GPCG [32], developed for strictly convex BQPs. It alternates between two phases: an identification phase, which performs GP iterations until a suitable face of the feasible set is identified or no reasonable progress toward the solution is achieved, and a minimization phase, which uses the ...

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On the Identification Property of a Projected Gradient Method

Cited by 9 publications

References 18 publications

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

On the regularizing behavior of the SDA and SDC gradient methods in the solution of linear ill-posed problems

A Two-Phase Gradient Method for Quadratic Programming Problems with a Single Linear Constraint and Bounds on the Variables

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