A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces

Boţ, Radu Ioan; Wanka, Gert

doi:10.1016/j.na.2005.09.017

Cited by 75 publications

(60 citation statements)

References 16 publications

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“…Given a subset U of X , by cl U we denote its closure in the corresponding topology, while its indicator function δ U : X → R = R ∪ {±∞} and respectively support function σ U : X * → R are defined as follows: We define in the following a notion that extends the one of a closed set, needed for being able to provide generalized closedness type characterizations via epigraphs for ε-duality gap statements, extending thus the investigations begun in [11,12] and continued later in many works such as [2, 5-9, 13, 15-17]. Note that the notion of an ε-closed set was considered in the literature in different instances that have nothing in common with our research, see for instance [1,14], while in [18,Definition 3.2] one can find the definition of a vertically closed set.…”

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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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confidence: 99%

Characterizations of ɛ-duality gap statements for constrained optimization problems

Boncea

Grad

2013

Open Mathematics

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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%

“…We can find conditions involving the continuity of one of the two functions at a point of dom f ∩ dom g, and some classical generalized interior point ones [1,14,28]. Recently, it has been introduced regularity conditions involving lower semicontinuity of the two functions, named closedness-type regularity conditions [5,6], which are not comparable with the most recent point-interior ones in the infinite dimensional setting (see [2]). In the e-convex context, the conditions obtained in this paper for the fulfilment of strong Fenchel duality can be regarded as the closedness-type ones in the classical convex setting, hence the problem of obtaining interiority-type conditions and the comparison of both type is a very interesting problem to work on in a future paper.…”

Section: Proof (I) ⇒ (Iimentioning

confidence: 99%

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

Fajardo

Vicente-Pérez

Rodríguez

2011

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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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“…The following notion of Cartesian product map is used in [7]: Definition 2. For the whole paper, we endow X * × R with the product topology of w * (X * , X) and the usual Euclidean topology.…”

Section: G(z) + H(x − Z) Note That If G H Is Exact At X Then (G H)(mentioning

confidence: 99%

“…Part (i) is a consequence of the RockafellarMoreau theorem (cf. [18,20]), and as in [7 3. New regularity conditions.…”

Section: G(z) + H(x − Z) Note That If G H Is Exact At X Then (G H)(mentioning

confidence: 99%

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

Li¹,

Fang²,

López-Acedo³

et al. 2009

SIAM J. Optim.

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Abstract. We consider the optimization problem (P A ) inf x∈X {f (x) + g(Ax)} where f and g are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y , respectively, and A is a linear operator from X to Y . By using the properties of the epigraph of the conjugated functions, some sufficient and necessary conditions for the strong Fenchel duality and the strong converse Fenchel duality of (P A ) are provided. Sufficient and necessary conditions for the stable Fenchel duality and for the total Fenchel duality are also derived.

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A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces

Cited by 75 publications

References 16 publications

Characterizations of ɛ-duality gap statements for constrained optimization problems

Characterizations of ɛ-duality gap statements for constrained optimization problems

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

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

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