Augmented Lagrangian Duality for Composite Optimization Problems

Kan, Chao; Song, Wen

doi:10.1007/s10957-014-0640-5

Cited by 7 publications

(12 citation statements)

References 18 publications

(51 reference statements)

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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%

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%

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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“…where [·] + denotes the projection of a matrix onto the cone of m × m positive semidefinite matrices. In order to obtain necessary and sufficient conditions for the existence of an augmented Lagrange multiplier for the problem (22), let us recall KKT optimality conditions for this problem [35,41]. Let x * be a locally optimal solution of the problem (22), and the functions f 0 , G and h be twice differentiable at x * .…”

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

“…Let x * be a locally optimal solution of the problem (22), and the functions f 0 , G and h be twice differentiable at x * . A pair (x * , λ * ), where λ * = (µ * , ν * ) ∈ Λ, is called a KKT pair of the problem (22), if µ * is positive semidefinite, µ * G(x * ) = 0 and D x L(x * , λ * ) = 0, where L(x, λ) = f 0 (x) + µ • G(x) + ν T h(x) is the classical Lagrangian. Suppose that rank(G(x * )) < m. One says that a KKT pair (x * , λ * ) satisfies the second order sufficient optimality condition, if the matrix…”

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

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Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

Долгополик

2017

Math. Program.

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In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near globally optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.

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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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Augmented Lagrangian Duality for Composite Optimization Problems

Cited by 7 publications

References 18 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

Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

Minimax exactness and global saddle points of nonlinear augmented Lagrangians

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