Augmented Lagrangian Algorithms Based on the Spectral Projected Gradient Method for Solving Nonlinear Programming Problems

Diniz-Ehrhardt, M. A.; Gomes-Ruggiero, Márcia A.; Martínez, José Mário; Santos, Sandra A.

doi:10.1007/s10957-004-5720-5

Cited by 29 publications

(17 citation statements)

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“…Clearly, any infeasible stationary point of (8) must be degenerate. Moreover, if x is infeasible and degenerate, by (13) and the KKT conditions of (12), the gradients of the equality constraints and the active bound constraints are linearly dependent. The reciprocal is not true.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

et al. 2006

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Abstract. Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.

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Section: ⊓ ⊔mentioning

confidence: 99%

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

et al. 2006

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“…( A(x) is positive semidefinite.) Moreover, following the spectral gradient philosophy [4,7,8,15,16,18,24,39,43,44] we impose:…”

Section: Description Of the Methodsmentioning

confidence: 99%

Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization

Birgin

Martínez

2007

Comput Optim Appl

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Augmented Lagrangian methods for large-scale optimization usually require efficient algorithms for minimization with box constraints. On the other hand, active-set box-constraint methods employ unconstrained optimization algorithms for minimization inside the faces of the box. Several approaches may be employed for computing internal search directions in the large-scale case. In this paper a minimal-memory quasi-Newton approach with secant preconditioners is proposed, taking into account the structure of Augmented Lagrangians that come from the popular Powell-Hestenes-Rockafellar scheme. A combined algorithm, that uses the quasi-Newton formula or a truncated-Newton procedure, depending on the presence of active constraints in the penalty-Lagrangian function, is also suggested. Numerical experiments using the Cute collection are presented.

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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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Augmented Lagrangian Algorithms Based on the Spectral Projected Gradient Method for Solving Nonlinear Programming Problems

Cited by 29 publications

References 1 publication

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization

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

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