On the Convergence of Augmented Lagrangian Methods for Constrained Global Optimization

Luo, Hezhi; Sun, Xiaoling; Li, Duan

doi:10.1137/060667086

Cited by 38 publications

(45 citation statements)

References 43 publications

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“…There exist many global optimization techniques for nonlinear programming problems, e.g., [2,3,4,9,18,20,21,22,23,24,28,29,31,38,43,44,48,49,50,51,52,53,58]. However, up to our knowledge, the proposed approach is the first practical deterministic global optimization method based on the Augmented Lagrangian framework.…”

Section: Introductionmentioning

confidence: 99%

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

2009

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A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the ε k -global minimization of the Augmented Lagrangian with simple constraints, where ε k → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

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Section: Introductionmentioning

confidence: 99%

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

2009

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“…The method dates back to [19,26,27,28], and had been further developed and studied from various angles in [33,3,10,20,13,9,32,5,1,2,6,7,23], among extensive other literature (see also the monographs [4,25,12,31]). Successful software based on the augmented Lagrangians includes LANCELOT [11] and ALGENCAN [34].…”

Section: Introduction Given Twice Continuously Differentiable Functimentioning

confidence: 99%

Local Convergence of Exact and Inexact Augmented Lagrangian Methods under the Second-Order Sufficient Optimality Condition

Fernández¹,

Solodov²

2012

SIAM J. Optim.

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Abstract. We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the second-order sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second-order sufficient condition. That said, the classical results allow the initial multiplier estimate to be far from the optimal one, at the expense of proportionally increasing the threshold value for the penalty parameters. Although our primary goal is to avoid constraint qualifications, if the stronger assumptions are introduced, then starting points far from the optimal multiplier are allowed within our analysis as well. Using only the second-order sufficient optimality condition, for penalty parameters large enough we prove primal-dual Q-linear convergence rate, which becomes superlinear if the parameters are allowed to go to infinity. Both exact and inexact solutions of subproblems are considered. In the exact case, we further show that the primal convergence rate is of the same Q-order as the primal-dual rate. Previous assertions for the primal sequence all had to do with the weaker R-rate of convergence and required the stronger assumptions cited above. Finally, we show that under our assumptions one of the popular rules of controlling the penalty parameters ensures their boundedness.

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“…The strong duality properties and exact penalization related to different types of nonlinear Lagrangians have been studied by many scholars (see e.g., [14,7,28,33,34]). The convergence of augmented Lagrangian methods for nonconvex constrained problems have been studied in [1,2,[4][5][6]11,[18][19][20][21][22]27,36].…”

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Saddle points of general augmented Lagrangians for constrained nonconvex optimization

Luo

2011

J Glob Optim

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Local and global saddle point conditions for a general augmented Lagrangian function proposed by Mangasarian are investigated in the paper for inequality and equality constrained nonconvex optimization problems. Under second order sufficiency conditions, it is proved that the augmented Lagrangian admits a local saddle point, but without requiring the strict complementarity condition. The existence of a global saddle point is then obtained under additional assumptions that do not require the compactness of the feasible set and the uniqueness of global solution of the original problem.

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On the Convergence of Augmented Lagrangian Methods for Constrained Global Optimization

Cited by 38 publications

References 43 publications

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

Local Convergence of Exact and Inexact Augmented Lagrangian Methods under the Second-Order Sufficient Optimality Condition

Saddle points of general augmented Lagrangians for constrained nonconvex optimization

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