Subdifferential of the conjugate function in general Banach spaces

Correa, Fernando; Hantoute, Abderrahim

doi:10.1007/s11750-011-0238-0

Cited by 6 publications

(8 citation statements)

References 7 publications

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“…First, in this section, we apply the previous results to extend the classical Fenchel duality to the nonconvex framework. This will lead us to recover some of the results in [3,4,5] (see, also, [26]), relating the solution set of a nonconvex optimization problem and its convexified relaxation. Second, we establish Fritz-John and KKT optimality conditions for convex semi-infinite optimization problems, improving similar results in [8].…”

Section: Two Applications In Optimizationmentioning

confidence: 67%

“…Proof Following similar arguments as those used in [4], we apply Proposition 14 in the duality pair ((X * * , w * * ), (X * , * )), replacing the function g in (69) by the functionĝ defined on X * * aŝ g(y) = g(y), if y ∈ X * * ; +∞, otherwise .…”

Section: Two Applications In Optimizationmentioning

confidence: 99%

“…The main results of this paper are applied to derive formulas for the subdifferential of the conjugate function ( [3], [4], [5]). Our approach permits simple proofs of these results, with the aim of relating the solution set of a nonconvex optimization problem and its convexified relaxation.…”

Section: Introductionmentioning

confidence: 99%

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Subdifferential of the Supremum via Compactification of the Index Set

2020

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In this paper we develop general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the noncontinuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semicontinuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming. Keywords Supremum of convex functions • subdifferentials • Stone-Čech compactification • convex semi-infinite programming • Fritz-John and KKT optimality conditions Mathematics Subject Classification (2010) 46N10 • 52A41 • 90C25

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Section: Two Applications In Optimizationmentioning

confidence: 67%

Section: Two Applications In Optimizationmentioning

confidence: 99%

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Subdifferential of the Supremum via Compactification of the Index Set

2020

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“…Since it can be shown that convolution in the equality of Theorem 8 is exact, we see that Theorem 8 corresponds to the set equality (7) co It is well known that when the inf-convolution is exact, then its epigraph is the sum of the epigraphs of the two functions.…”

Section: Introduction Saint Raymond Observesmentioning

confidence: 75%

On the Klee--Saint Raymond's Characterization of Convexity

Corrêa¹,

Hantoute²,

Pérez-Aros³

2016

SIAM J. Optim.

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“…The main results of this paper are applied to derive formulas for the subdifferential of the conjugate function [3][4][5]23]. Our approach permits simple proofs of these results, with the aim of relating the solution set of a nonconvex optimization problem and its convexified relaxation.…”

Section: Introductionmentioning

confidence: 99%

Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

2020

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In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.

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Subdifferential of the conjugate function in general Banach spaces

Cited by 6 publications

References 7 publications

Subdifferential of the Supremum via Compactification of the Index Set

Subdifferential of the Supremum via Compactification of the Index Set

On the Klee--Saint Raymond's Characterization of Convexity

Subdifferential of the supremum function: moving back and forth between continuous and non-continuous settings

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