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Abderrahim Hantoute

Publications and source records attributed to Abderrahim Hantoute.

16 recordsLinked to original sources
arXiv2026

Explicit data-dependent characterizations of the subdifferential of convex pointwise suprema and optimality conditions

We establish explicit data-dependent and symmetric characterizations of the subdifferential of the supremum of convex functions, formulated directly in terms of the underlying data functions. In our approach, both active and non-active functions contribute equally through their subdifferentials, thereby avoiding the need for additional geometric constructions, such as the domain of the supremum, that arise in previous developments. Applications to infinite convex optimization yield sharp Karush-Kuhn-Tucker and Fritz-John optimality conditions, expressed exclusively in terms of the objective and constraint functions and clearly distinguishing the roles of (almost) active and non-active constraints.

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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

Strong duality in infinite convex optimization

We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums of functions. Unlike the classical Haar duality scheme, these dual problems provide zero duality gap and are solvable under the standard Slater condition. Then we derive general optimality conditions/multiplier rules by applying subdifferential rules for infinite sums established in [13].

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arXiv2025

Optimality conditions and subdifferential calculus for infinite sums of functions

The paper extends the widely used in optimisation theory decoupling techniques to infinite collections of functions. Extended concepts of uniform lower semicontinuity and firm uniform lower semicontinuity are discussed. The main theorems give fuzzy subdifferential necessary conditions (multiplier rules) for a local minimum of the sum of an infinite collection of functions and fuzzy subdifferential sum rules without the traditional Lipschitz continuity assumptions. More subtle "quasi" versions of the uniform infimum and uniform lower semicontinuity properties are also discussed.

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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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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

Coerciveness condition for quasi-equilibrium problems

A quasi-equilibrium problem is an equilibrium problem where the constraint set does depend on the reference point. It generalizes important problems such as quasi-variational inequalities and generalized Nash equilibrium problems. We study the existence of equilibria on unbounded sets under a coerciveness condition adapted from one specific for quasi-variational inequalities recently proposed by Aussel and Sultana. We discuss the relation of our results with others that are present in the literature.

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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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arXiv2018

Equivalence between Differential Inclusions Involving Prox-regular sets and maximal monotone operators

In this paper, we study the existence and the stability in the sense of Lyapunov of solutions for\ differential inclusions governed by the normal cone to a prox-regular set and subject to a Lipschitzian perturbation. We prove that such, apparently, more general nonsmooth dynamics can be indeed remodelled into the classical theory of differential inclusions involving maximal monotone operators. This result is new in the literature and permits us to make use of the rich and abundant achievements in this class of monotone operators to derive the desired existence result and stability analysis, as well as the continuity and differentiability properties of the solutions. This going back and forth between these two models of differential inclusions is made possible thanks to a viability result for maximal monotone operators. As an application, we study a Luenberger-like observer, which is shown to converge exponentially to the actual state when the initial value of the state's estimation remains in a neighborhood of the initial value of the original system.

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arXiv2017

A general representation of delta-normal sets to sublevels of convex functions

The (delta-) normal cone to an arbitrary intersection of sublevel sets of proper, lower semicontinuous, and convex functions is characterized, using either epsilon-subdifferentials at the nominal point or exact subdifferentials at nearby points. Our tools include (epsilon-) calculus rules for sup/max functions. The framework of this work is that of a locally convex space, however, formulas using exact subdifferentials require some restriction either on the space (e.g. Banach), or on the function (e.g. epi-pointed).

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arXiv2017

Subdifferential characterization of probability functions under Gaussian distribution

Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth.This fact motivates the consideration of subdifferentials for such typically just continuous functions. The aim of this paper is to provide subdifferential formulae in the case of Gaussian distributions for possibly infinite-dimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the spheric-radial decomposition of Gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other. By successively adding additional hypotheses, conditions are satisfied under which the probability function is locally Lipschitzian or even differentiable.

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arXiv2013

Local nonsmooth Lyapunov pairs for first-order evolution differential inclusions

The general theory of Lyapunov's stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in a previous work. This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior. This setting permits to have nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion. It also allows some more explicit criteria for Lyapunov's pairs. Some consequences to the viability of closed sets are given, as well as some useful cases relying on the continuity or/and convexity of the involved functions. Our analysis makes use of standard tools from convex and variational analysis.

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