Duality and Exact Penalization for General Augmented Lagrangians

Burachik, Regina S.; Iusem, Alfredo N.; Melo, Jefferson G.

doi:10.1007/s10957-010-9711-4

Cited by 24 publications

(43 citation statements)

References 16 publications

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“…r ≥ 0, and r(λ) = 1 + |λ| for any λ ∈ Λ. Therefore r(λ) > r(0) for any λ = 0, where r(0) is, in fact, the least exact penalty parameter of the ℓ 1 penalty function for the problem (8).…”

Section: Reduction To Exact Penalty Functionsmentioning

confidence: 98%

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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Section: Reduction To Exact Penalty Functionsmentioning

confidence: 98%

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

Долгополик

2017

Math. Program.

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“…Proof. Our aim is to apply the localization principle in the parametric form to the separating function (14). To this end, define G(·) = (g 1 (·), .…”

Section: Example Iii: Continuously Differentiable Exact Penalty Functmentioning

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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“…The dual problem generated by such Lagrangians as in (1) is a (nondifferentiable) convex problem, which is usually solved by nonsmooth convex optimization techniques, such as subgradient methods and their extensions [2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

Freire

Lemonge

Fonseca

et al. 2018

Mathematical Problems in Engineering

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The Interior Epigraph Directions (IED) method for solving constrained nonsmooth and nonconvex optimization problem via Generalized Augmented Lagrangian Duality considers the dual problem induced by a Generalized Augmented Lagrangian Duality scheme and obtains the primal solution by generating a sequence of iterates in the interior of the epigraph of the dual function. In this approach, the value of the dual function at some point in the dual space is given by minimizing the Lagrangian. The first version of the IED method uses the Matlab routine fminsearch for this minimization. The second version uses NFDNA, a tailored algorithm for unconstrained, nonsmooth and nonconvex problems. However, the results obtained with fminsearch and NFDNA were not satisfactory. The current version of the IED method, presented in this work, employs a Genetic Algorithm, which is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as GA, is applied alone to solve constrained optimization problems. Two sets of constrained optimization problems from mathematics and mechanical engineering were solved and compared with literature. It is shown that the proposed hybrid algorithm is able to solve problems where fminsearch and NFDNA fail.

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Duality and Exact Penalization for General Augmented Lagrangians

Cited by 24 publications

References 16 publications

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

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

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

A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

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