New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces

Boţ, Radu Ioan; Grad, Sorin‐Mihai; Wanka, Gert

doi:10.1016/j.na.2007.05.021

Cited by 55 publications

(25 citation statements)

References 9 publications

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“…[3][4][5]) in the direction of ε-vertical closedness can be considered, too. Note that one can formulate also ε-optimality conditions statements for (P) and its considered dual problems, extending thus the corresponding optimality conditions statements from [6,7]. Moreover, ε-Farkas type results can be given for the considered problem by combining the statements from this paper with ideas from [10].…”

Section: Conclusion and Further Researchmentioning

confidence: 96%

“…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%

“…Motivated by the characterizations of the stable strong duality from [6,7] we begin this section with several equivalent representations of various instances of ε-duality gap for (P) and its duals by means of epigraphs.…”

Section: ε-Duality Gap Statements Involving Epigraphs For Lagrange Anmentioning

confidence: 99%

“…Taking ε = 0, Theorem 3.1 becomes [6,Theorem 5] without the topological and convexity assumptions on the involved functions, Corollary 3.2 is [6, Theorem 6], while Corollary 3.3 turns into [7,Theorem 3]. Note also that by considering (RCL) for ( ) = 0 whenever ∈ X one can extend (analogously to Theorem 3.12, see also Remarks 3.13 and 3.14) [6, Theorem 9] towards ε-duality gap and the same can be done for their counterparts corresponding to the other considered duals.…”

Section: Remark 35mentioning

confidence: 99%

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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: Conclusion and Further Researchmentioning

confidence: 96%

Section: Preliminariesmentioning

confidence: 99%

Section: ε-Duality Gap Statements Involving Epigraphs For Lagrange Anmentioning

confidence: 99%

Section: Remark 35mentioning

confidence: 99%

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Characterizations of ɛ-duality gap statements for constrained optimization problems

Boncea

Grad

2013

Open Mathematics

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“…In the classical context, with f and g t for all t 2 T proper convex and lsc functions, [1] showed that stable strong Lagrange duality is equivalent to the closedness of the set S…”

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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Metric and Topological Tools

Penot

2012

Calculus Without Derivatives

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New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces

Cited by 55 publications

References 9 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

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