Spectral Projected Gradient Methods: Review and Perspectives

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

doi:10.18637/jss.v060.i03

Cited by 142 publications

(126 citation statements)

References 106 publications

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“…They introduce a new globalization approach combining nonmonotone watchdog technique with line search rules. A detail survey on Barzilai-Borwein method-review and perspectives can be found in [19].…”

Section: An Iterative Methods For Solving the Discrete Problemmentioning

confidence: 99%

Nonlocal problem for a general second-order elliptic operator

Todorov

2015

Computers & Mathematics with Applications

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a b s t r a c tThe paper deals with a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the strong problem essentially increases its complexity and the necessary total computational work. The existence and uniqueness of the weak solution is established. The nonlinear weak formulation is reduced to the minimization of a nonlinear functional. Finite element discretizations by Lagrangian finite elements are applied to obtain an approximate minimization problem. A two-point step size gradient method with an original steplength is used for finding approximate solutions of the problem under consideration. No line search is necessary in the new approach. The present method is computer implemented and tested on different triangulations. The test examples indicate that the method slightly depends on the initial guesses.

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Section: An Iterative Methods For Solving the Discrete Problemmentioning

confidence: 99%

Nonlocal problem for a general second-order elliptic operator

Todorov

2015

Computers & Mathematics with Applications

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“…Due to its simplicity and numerical efficiency, the spectral gradient as well as the spectral projected gradient method has been applied successfully to finding local minimizers of large scale problems [3,4,7,11,14,15,19,27], for more details, see the recent paper [5] and references therein.…”

Section: Then the Secant Equation Becomesmentioning

confidence: 99%

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Yu¹,

Gan²

2016

J. Nonlinear Sci. Appl.

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In this paper, we propose a spectral projected gradient method for the convex optimization problem with singular solution. By solving the equivalent equation of the gradient function, this method combines the perturbed spectral gradient direction with the projection direction to generate the next iteration point. Under some mild conditions, we establish the global convergence and the local R-linear convergence rate under the local error bound condition. Preliminary numerical tests are given to show that the proposed method works well.

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“…Recently, the BB method and its variants have been successfully extended to general unconstrained problems [35], to constrained optimization problems [3,28] and to various applications [29,30,32]. One may see [4,10,17,21,37] and the references therein. Let {ξ 1 , ξ 2 , .…”

Section: Introductionmentioning

confidence: 99%

Gradient methods exploiting spectral properties

Huang

Dai

et al. 2020

Optimization Methods and Software

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We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73-88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop a monotone gradient method that takes a certain number of steps using the asymptotically optimal stepsize by Dai and Yang, and then follows by some short steps associated with this new stepsize. By employing one step retard of the asymptotic optimal stepsize, a nonmonotone variant of this method is also proposed. Under mild conditions, R-linear convergence of the proposed methods is established for minimizing quadratic functions. In addition, by combining gradient projection techniques and adaptive nonmonotone line search, we further extend those methods for general bound constrained optimization. Two variants of gradient projection methods combining with the Barzilai-Borwein stepsizes are also proposed. Our numerical experiments on both quadratic and bound constrained optimization indicate that the new proposed strategies and methods are very effective.

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Spectral Projected Gradient Methods: Review and Perspectives

Cited by 142 publications

References 106 publications

Nonlocal problem for a general second-order elliptic operator

Nonlocal problem for a general second-order elliptic operator

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Gradient methods exploiting spectral properties

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