Stable and Total Fenchel Duality for Convex Optimization Problems in Locally Convex Spaces

Li, Chong; Fang, Dingzhi; López-Acedo, Genaro; López, Marco A.

doi:10.1137/080734352

Cited by 36 publications

(21 citation statements)

References 13 publications

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“…Motivated by recent results on stable strong and total duality for constrained convex optimization problems in [2,6,7,9,13,17] and the ones on zero duality gap in [15,16] we introduce in this paper several regularity conditions which characterize ε-duality gap statements (with ε ≥ 0) for a constrained optimization problem and its Lagrange and FenchelLagrange dual problems, respectively. The regularity conditions we provide in Section 2 are based on epigraphs, while the ones in Section 3 on ε-subdifferentials.…”

Section: Preliminariesmentioning

confidence: 99%

Characterizations of ɛ-duality gap statements for constrained optimization problems

Boncea

Grad

2013

Open Mathematics

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In this paper we present different regularity conditions that equivalently characterize various ε-duality gap statements (with ε ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ε-subdifferentials. When ε = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature. MSC: PreliminariesMotivated by recent results on stable strong and total duality for constrained convex optimization problems in [2,6,7,9,13,17] and the ones on zero duality gap in [15,16] we introduce in this paper several regularity conditions which characterize ε-duality gap statements (with ε ≥ 0) for a constrained optimization problem and its Lagrange and FenchelLagrange dual problems, respectively. The regularity conditions we provide in Section 2 are based on epigraphs, while the ones in Section 3 on ε-subdifferentials. In this way we extend many of the results in the mentioned papers, which are recovered as special cases when ε = 0, delivering thus generalizations of the classical Farkas-Minkowski and basic constraint qualifications. Moreover some statements in [6-8, 15, 16], which arise from our results in the special *

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Section: Preliminariesmentioning

confidence: 99%

Characterizations of ɛ-duality gap statements for constrained optimization problems

Boncea

Grad

2013

Open Mathematics

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“…Our characterization is motivated by [11], where in the classical setting, strong Fenchel duality is equivalent to the inequality (f + A ) (0) (f A ) (0) ; together with the exactness of the infimal convolution at the point 0; being f and A proper and convex functions. In [6] it was shown that, if v (P ) 2 R and f + A = sup n a j a 2 e E f; A o ; condition (C3) is sufficient for strong Fenchel duality.…”

Section: Proposition 41 Condition (C F ) Guarantees Stable Strong Fementioning

confidence: 99%

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

Fajardo

Vidal-Núñez

2016

Optimization

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By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for disturbed primal problems by continuous linear functionals and their respective dual problems, which is named stable strong duality, are stablished. In these conditions, the evenly convexity of the perturbation function will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal-dual problems, obtaining a characterization for Fenchel case.

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“…In the convex setting with finite dimensional space, background information on Fenchel duality is due to Rockafellar [26,27]. With infinite dimensional space, we refer for instance to [5,7,11,23,17].…”

Section: Proof (I) ⇒ (Iimentioning

confidence: 99%

Infimal convolution, c-subdifferentiability, and Fenchel duality in evenly convex optimization

Fajardo

Vicente-Pérez

Rodríguez

2011

TOP

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In this paper we deal with strong Fenchel duality for infinite dimensional optimization problems where both feasible set and objective function are evenly convex. To this aim, via perturbation approach, a conjugation scheme for evenly convex functions, based on generalized convex conjugation, is used. The key is to extend some well-known results from convex analysis, involving the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of ε-subdifferentials for lower semicontinuous convex functions, to this more general framework.Mathematical subject classification: 52A20, 26B25.

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Stable and Total Fenchel Duality for Convex Optimization Problems in Locally Convex Spaces

Cited by 36 publications

References 13 publications

Characterizations of ɛ-duality gap statements for constrained optimization problems

Characterizations of ɛ-duality gap statements for constrained optimization problems

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

Infimal convolution, c-subdifferentiability, and Fenchel duality in evenly convex optimization

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