The e-support function of an e-convex set and conjugacy for e-convex functions

Martínez-Legaz, Juan Enrique; Vicente-Pérez, José

doi:10.1016/j.jmaa.2010.10.058

Cited by 20 publications

(44 citation statements)

References 4 publications

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“…Based on the generalized convex conjugation theory introduced by Moreau [16], a suitable conjugation scheme is provided in [15] for e-convex functions. Consider the set W := X X R with the coupling functions c : X W !…”

Section: Is a Proper E-convex Function If And Only Ifmentioning

confidence: 99%

“…c(x; (x ; y ; )) 2 R, with (x ; y ; ) 2 W and 2 R are called c-elementary; in the same way, c 0 -elementary functions are those of the form (x ; y ; ) 2 W ! c(x; (x ; y ; )) 2 R, with x 2 X and 2 R. In [15] it is shown that the family of the proper e-convex functions from X to R along with the function identically equal to 1 is actually the family of pointwise suprema of sets of celementary functions. Using an analogous terminology, a function g : W !…”

Section: Is a Proper E-convex Function If And Only Ifmentioning

confidence: 99%

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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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Section: Is a Proper E-convex Function If And Only Ifmentioning

confidence: 99%

Section: Is a Proper E-convex Function If And Only Ifmentioning

confidence: 99%

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

Fajardo

Vidal-Núñez

2016

Optimization

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“…For a function h : X → R we denote Δh := eco(dom h) ⊂ X, a set of crucial importance as observed in [11,15]. …”

Section: It Holds That Epi(co H) = Co(epi H) H γ ≤ Co H ≤ Co H ≤ H Amentioning

confidence: 99%

“…Recently, a Moreau generalized conjugacy related to evenly convex functions has been introduced in [11]. Another approach, using evenly quasiconvex duality and the conjugacy machinery (see [9,14,18], for instance), is given in [19].…”

Section: Evenly Convex Conjugacy In General Topological Vector Spacesmentioning

confidence: 99%

“…Previously, in the eighties, evenly quasiconvex functions (those with evenly convex sublevel sets) were introduced in quasiconvex programming [9,14]. Although the definitions of evenly convex set and evenly convex function were given in a finite-dimensional space, they have been also considered in any separated locally convex space [4,11,19] in a natural way.…”

Section: Introductionmentioning

confidence: 99%

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Even convexity, subdifferentiability, and Γ-regularization in general topological vector spaces

Vicente-Pérez

Volle

2015

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: J. Vicente-Pérez, M. Volle, Even convexity, subdifferentiability, and Γ-regularization in general topological vector spaces, J. Math. Anal. Appl. (2015), http://dx.doi.org/10. 1016/j.jmaa.2015.04.051 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.Even convexity, subdifferentiability, and Γ-regularization in general topological vector spacesRevised version: April 15, 2015 AbstractIn this paper we provide new results on even convexity and extend some others to the framework of general topological vector spaces. We first present a characterization of the even convexity of an extended real-valued function at a point. We then establish the links between even convexity and subdifferentiability and the Γ-regularization of a given function. Consequently, we derive a sufficient condition for strong duality fulfillment in convex optimization problems.

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