arXiv subjects

Rafael Correa

Publications and source records attributed to Rafael Correa.

9 recordsLinked to original sources

arXiv2026

Normal cones to sublevel sets of convex and quasi-convex supremum functions

We provide sharp and explicit characterizations of the normal cone to sublevel sets of suprema of arbitrary functions, expressed exclusively in terms of subdifferentials of the data functions. In the convex case, the resulting formulas involve the approximate subdifferential of the individual data functions at the nominal point. In contrast, the quasi-convex framework requires the use of the Fr\'echet subdifferential of these data functions but evaluated at nearby points. These results are applied to derive optimality conditions for infinite convex and quasi-convex optimization problems.

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arXiv2026

Relaxation in infinite convex programming under Slater-type regularity conditions

The main purpose of this paper is to close the gap between the optimal values of an infinite convex program and that of its biconjugate relaxation. It is shown that Slater and continuity-type conditions guarantee such a zero-duality gap. The approach uses calculus rules for the conjugation and biconjugation of the sum and pointwise supremum operations. A second important objective of this work is to exploit these results on relaxation by applying them in the context of duality theory.

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arXiv2025

Duality for the $\Gamma$-convergence of convex functions

We extend the duality principle for the $\Gamma$-convergence of convex lower semicontinuous functions, which was previously established only in separable reflexive Banach spaces, to the broader class of weakly compactly generated (WCG) Banach spaces, addressing a question of Fitzpatrick and Lewis. Under the same classical hypothesis of equicoercivity, we show that $\Gamma$-convergence in the norm topology is equivalent to $\Gamma$-convergence of the Fenchel conjugates in the weak$^\ast$ topology. We further prove that this duality is equivalent to the graphical convergence of the associated subdifferentials with respect to the product topology given by the norm on the primal space and the weak$^\ast$ topology on the dual. The WCG setting encompasses all separable and all reflexive Banach spaces separately, i.e, separable spaces without reflexivity assumptions and reflexive spaces without separability assumptions, as well as important non-reflexive spaces which may fail to be separable, such as $L^1(\mu)$ for an arbitrary $\sigma$-finite measure. As an application, we derive dual characterizations of the $\Gamma$-convergence of convex integral functionals on $L^p$ spaces ($1\leq p<\infty $).

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arXiv2024

Convex regularization and subdifferential calculus

This paper deals with the regularization of the sum of functions defined on a locally convex spaces through their closed-convex hulls in the bidual space. Different conditions guaranteeing that the closed-convex hull of the sum is the sum of the corresponding closed-convex hulls are provided. These conditions are expressed in terms of some epsilon-subdifferential calculus rules for the sum. The case of convex functions is also studied, and exact calculus rules are given under additional continuity/qualifications conditions. As an illustration, a variant of the proof of the classical Rockafellar theorem on convex integration is proposed.

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arXiv2023

Optimality conditions in DC constrained mathematical programming problems

This paper provides necessary and sufficient optimality conditions for abstract constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections.

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arXiv2020

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

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 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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arXiv2019

Qualification conditions-free characterizations of the $\varepsilon$-subdifferential of convex integral functions

We provide formulae for the $\varepsilon$-subdifferential of the integral function $ I_f(x):=\int_T f(t,x) dμ(t), $ where the integrand $f:T\times X \to [-\infty,+\infty]$ is measurable in $(t,x)$ and convex in $x$. The state variable lies in a locally convex space, possibly non-separable, while $T$ is given a structure of a nonnegative complete $σ$-finite measure space $(T,\mathcal{A},μ)$. The resulting characterizations are given in terms of the $ε$-subdifferential of the data functions involved in the integrand, $f$, without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.

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arXiv2019

Characterizations of the subdifferential of convex integral functions under qualification conditions

This work provides formulae for the $ε$-subdifferential of integral functions in the framework of complete $σ$-finite measure spaces and locally convex spaces. In this work we present here new formulae for this $ε$-subdifferential under the presence of continuity-type qualification conditions relying on the data involved in the integrand. the integral function $I_f$ are also presented.

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arXiv2019

Sequential and exact formulae for the subdifferential of nonconvex integral functionals

This work concerns the study of the subdifferential of the integral functional $$ E_f(x)=\int_{T} f(t,x)dμ(t), $$ where $f$ is a (not necessarily convex) normal integrand, $({T},\mathcal{A},μ)$ is a $σ$-finite measure space, while the decision variables vary in a separable Asplund space. First, using techniques of variational analysis we establish sequential approximate formulae for the Fréchet subdifferential of $E_f$. Secondly, we introduce a Lipschitz-like condition, which allows us to give an upper-estimation for the limiting subdifferential of $E_{f}$ even when this functional is non-Lipschitz.

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