Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

Kan, Chao; Song, Wen

doi:10.1007/s10898-015-0273-8

Cited by 6 publications

(4 citation statements)

References 31 publications

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“…In order to illuminate the main features of parametric exactness, in this example we consider a separating function that depends on additional parameters, namely Lagrange multipliers. Below, we apply the general theory of parametrically exact separating functions to the augmented Lagrangian function introduced by Rockafellar and Wets in [92] (see also [100,61,62,131,40,99,66,67,16]). Let P be a topological vector space of parameters.…”

Section: Example Iv: Rockafellar-wets' Augmented Lagrangian Functionmentioning

confidence: 99%

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A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Долгополик¹

2018

J Optim Theory Appl

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In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications.In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

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Section: Example Iv: Rockafellar-wets' Augmented Lagrangian Functionmentioning

confidence: 99%

“…Finally, there is also a problem of the existence of augmented Lagrange multipliers, which can be viewed as the study of the global exactness of Rockafellar-Wets' augmented Lagrangian function. Various results on the existence of augmented Lagrange multipliers were obtained in [100,131,40,99,66,67,16].…”

Section: Introductionmentioning

confidence: 99%

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Долгополик¹

2018

J Optim Theory Appl

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“…This augmented Lagrangian was introduced in [17] and thoroughly analyzed by many researchers [18,19,20,21,22,23]. The existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian for various types of optimization problems was studied in [1,2,14,15,24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Minimax exactness and global saddle points of nonlinear augmented Lagrangians

2021

J. Appl. Numer. Optim.

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We study global minimax exactness of merit functions for constrained optimization problems. This concept arises as a natural generalization of the definition of global saddle points in the unified theory of exactness of penalty and augmented Lagrangian functions. We obtain necessary and sufficient conditions for the global minimax exactness of nonlinear augmented Lagrangians in the form of the localization principle, which allow one to reduce the study of the existence of global saddle points (or the existence of solutions of the augmented dual problem) to a local analysis of sufficient optimality conditions. With the use of the localization principle, we obtain simple necessary and sufficient conditions for the existence of global saddle points of He-Wu-Meng's augmented Lagrangian for inequality constrained problems and nonlinear rescaling Lagrangians for nonconvex semidefinite programs. We also introduce and analyze a new nonlinear smooth augmented Lagrangian for constrained minimax problems and provide necessary and sufficient conditions for the existence of a global saddle point of this augmented Lagrangian, which, in particular, expose some limitations of exponential penalty function methods.

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“…In addition, augmented Lagrange multipliers are closely related to saddle points, the zero duality gap property, and exact penalty representation. Some results on the existence of augmented Lagrange multipliers are discussed for semi-infinite programming [25], cone programming [22,23], and eigenvalue composite optimization problems [38]. Moreover, CQ-free duality was proposed in the classical monograph [39] by Bonnans and Shapiro.…”

Section: Introductionmentioning

confidence: 99%

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

Wang

Zhou

Tang

2020

MCA

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The augmented Lagrange multiplier as an important concept in duality theory for optimization problems is extended in this paper to generalized augmented Lagrange multipliers by allowing a nonlinear support for the augmented perturbation function. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis. Meanwhile, the relations among generalized augmented Lagrange multipliers, saddle points, and zero duality gap property are developed.

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Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

Cited by 6 publications

References 31 publications

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Minimax exactness and global saddle points of nonlinear augmented Lagrangians

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

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