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Michel Théra

Publications and source records attributed to Michel Théra.

18 recordsLinked to original sources
arXiv2026

Splitting Algorithms Using Pairs Monotonicity to Solve the Sum of Two Non-monotone Operators Problem

This paper is concerned with solving the problem of finding a zero of the sum of a single-valued operator and a set-valued operator, neither of which is assumed to be monotone, in a real Hilbert space. Under the framework of pair monotonicity, we propose a generalized forward--backward algorithm and a generalized Tseng algorithm based on transformed and warped resolvents associated with an auxiliary linear operator. We establish weak, strong and linear convergence of the proposed algorithms under suitable assumptions involving pair strong monotonicity, pair cocoercivity, and pair Lipschitz continuity. Finally, numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed algorithms.

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arXiv2026

Nonlinear and orbital directional regularities for set-valued mappings and applications to fixed points

Motivated by developments in metric regularity theory for set-valued mappings, this paper introduces several directional regularity notions that are weaker than the corresponding classical regularity properties. We investigate the relationships among these concepts and establish a classification framework based on directional minimal time functions. Particular attention is devoted to directional orbital regularity and directional orbital Aubin continuity, which are shown to provide a natural setting for the analysis of fixed point phenomena. Using these notions, we derive approximate and exact fixed point theorems for set-valued mappings under directional assumptions. The obtained results are expressed in terms of minimal time functions and orbital approximation procedures, allowing for anisotropic and asymmetric behaviors that are not captured by the classical nondirectional framework. We further apply the developed theory to directional coincidence point results, coupled fixed point theorems, and Milyutin-type perturbation stability properties. The proposed approach provides a unified directional perspective on fixed point theory and variational analysis. In particular, it recovers several known results from the literature as special cases and yields new extensions under substantially weaker assumptions than classical contraction or global regularity conditions.

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arXiv2026

Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators

This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To overcome these challenges, we introduce a regularized system based on a fixed Lipschitz approximation of the convexified mapping. From this approximation, we extract a single-valued Lipschitz selection that preserves the essential geometric features of the original system. This framework enables the application of nonsmooth Lyapunov methods and Hamiltonian-based stability criteria. Instead of approximating trajectories, we focus on analyzing a simplified system that faithfully reflects the structure of the original dynamics. Several examples are provided to illustrate the method's practicality and scope.

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arXiv2026

Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method

Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.

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arXiv2025

Weak and strong convergence of a relaxed inertial proximal splitting algorithm for solving hierarchical equilibrium problems

In this chapter, we introduce the relaxed inertial proximal splitting algorithm (RIPSA) for hierarchical equilibrium problems. Using Opial-Passty's lemma, we first establish weak ergodic and weak convergence of the sequence generated by the algorithm to a solution of the problem, in the absence of the Browder-Halpern contraction factor. We then derive a strong convergence result under an additional strong monotonicity assumption. Subsequently, we relax this requirement by removing strong monotonicity and instead incorporating a Browder-Halpern contraction factor into (RIPSA), which guarantees strong convergence to a solution determined by the contraction factor. Finally, we discuss two related settings: convex minimization problems and monotone variational inequalities formulated as fixed-point problems for nonexpansive operators.

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arXiv2025

An inertial proximal splitting algorithm for hierarchical bilevel equilibria in Hilbert spaces

In this article, we aim to approximate a solution to the bilevel equilibrium problem $\mathbf{(BEP})$ for short: find $\bar{x} \in \mathbf{S}_f$ such that $ g(\bar{x}, y) \geq 0, \,\, \forall y \in \mathbf{S}_f, $ where $ \mathbf{S}_f = \{ u \in \mathbf{K} : f(u, z) \geq 0, \forall z \in \mathbf{K} \}. $ Here, $\mathbf{K}$ is a closed convex subset of a real Hilbert space $\mathcal{H}$, and $f$ and $g$ are two real-valued bifunctions defined on $\mathbf{K} \times \mathbf{K}$. We propose an inertial version of the proximal splitting algorithm introduced by Z. Chbani and H. Riahi: \textit{Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems}. \textit{Numer. Algebra Control Optim.}, 3 (2013), pp. 353-366. Under suitable conditions, we establish the weak and strong convergence of the sequence generated by the proposed iterative method. We also report a numerical example illustrating our theoretical result.

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arXiv2025

Solving Non-Monotone Inclusions Using Monotonicity of Pairs of Operators

In this paper, under the monotonicity of pairs of operators, we propose some Generalized Proximal Point Algorithms to solve non-monotone inclusions using warped resolvents and transformed resolvents. The weak, strong, and linear convergence of the proposed algorithms are established under very mild conditions.

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arXiv2024

Subtransversality and Strong CHIP of Closed Sets in Asplund Spaces

In this paper, we mainly study subtransversality and two types of strong CHIP (given via Fréchet and limiting normal cones) for a collection of finitely many closed sets. We first prove characterizations of Asplund spaces in terms of subtransversality and intersection formulae of Fréchet normal cones. Several necessary conditions for subtransversality of closed sets are obtained via Fréchet/limiting normal cones in Asplund spaces. Then, we consider subtransversality for some special closed sets in convex-composite optimization. In this frame we prove an equivalence result on subtransversality, strong Fréchet CHIP and property (G) so as to extend a duality characterization of subtransversality of finitely many closed convex sets via strong CHIP and property (G) to the possibly non-convex case. As applications, we use these results on subtransversality and strong CHIP to study error bounds of inequality systems and give several dual criteria for error bounds via Fréchet normal cones and subdifferentials.

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arXiv2024

Perturbation Analysis of Error Bounds for Convex Functions on Banach Spaces

This paper focuses on the stability of both local and global error bounds for a proper lower semicontinuous convex function defined on a Banach space. Without relying on any dual space information, we first provide precise estimates of error bound moduli using directional derivatives. For a given proper lower semicontinuous convex function on a Banach space, we prove that the stability of local error bounds under small perturbations is equivalent to the directional derivative at a reference point having a non-zero minimum over the unit sphere. Additionally, the stability of global error bounds is shown to be equivalent to the infimum of the directional derivatives, at all points on the boundary of the solution set, being bounded away from zero over some neighborhood of the unit sphere.

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arXiv2024

Explicit Convergence Rate of The Proximal Point Algorithm under R-Continuity

The paper provides a thorough comparison between R-continuity and other fundamental tools in optimization such as metric regularity, metric subregularity and calmness. We show that R-continuity has some advantages in the convergence rate analysis of algorithms solving optimization problems. We also present some properties of R-continuity and study the explicit convergence rate of the Proximal Point Algorithm (PPA) under the R-continuity.

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arXiv2023

On Error Bounds of Inequalities in Asplund Spaces

Error bounds are central objects in optimization theory and its applications. They were for a long time restricted only to the theory before becoming over the course of time a field of itself. This paper is devoted to the study of error bounds of a general inequality defined by a proper lower semicontinuous function on an Asplund space. Even though the results of the dual characterization on the error bounds of a general inequality (if one drops the convexity assumption) may not be valid, several necessary dual conditions are still obtained in terms of Fréchet/Mordukhovich subdifferentials of the concerned function at points in the solution set. Moreover, for an inequality defined by a composite-convex function that is to say by a function which is the composition of a convex function with a smooth mapping, such dual conditions also turn out to be sufficient to have the error bound property. Our work is an extension of results on dual characterizations of convex inequalities to the possibly non-convex case.

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arXiv2023

Locating Theorems of Differential Inclusions Governed by Maximally Monotone Operators

In this paper, we are interested in studying the asymptotic behavior of the solutions of differential inclusions governed by maximally monotone operators. In the case where the LaSalle's invariance principle is inconclusive, we provide a refined version of the invariance principle theorem. This result derives from the problem of locating the $ω$-limit set of a bounded solution of the dynamic. In addition, we propose an extension of LaSalle's invariance principle, which allows us to give a sharper location of the $ω$-limit set. The provided results are given in terms of nonsmooth Lyapunov pair-type functions.

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arXiv2023

Primal Characterizations of Stability of Error Bounds for Semi-infinite Convex Constraint Systems in Banach Spaces

This article is devoted to the stability of error bounds (local and global) for semi-infinite convex constraint systems in Banach spaces. We provide primal characterizations of the stability of local and global error bounds when systems are subject to small perturbations. These characterizations are given in terms of the directional derivatives of the functions that enter into the definition of these systems. It is proved that the stability of error bounds is essentially equivalent to verifying that the optimal values of several minimax problems, defined in terms of the directional derivatives of the functions defining these systems, are outside of some neighborhood of zero. Moreover, such stability only requires that all component functions entering the system have the same linear perturbation. When these stability results are applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, primal criteria for Hoffman's constants to be uniformly bounded under perturbations of the problem data are obtained.

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arXiv2022

Weak and strong convergence of an inertial proximal method for solving bilevel monotone equilibrium problems

In this paper, we introduce an inertial proximal method for solving a bilevel problem involving two monotone equilibrium bifunctions in Hilbert spaces. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the iterative method are established. Two particular cases illustrating the proposed method are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under saddle point constraint. Furthermore, a numerical example is given to demonstrate the implementability of our algorithm. The algorithm and its convergence results improve and develop previous results in the field.

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arXiv2022

Primal Characterizations of Error Bounds for Composite-convex Inequalities

This paper is devoted to primal conditions of error bounds for a general function. In terms of Bouligand tangent cones, lower Hadamard directional derivatives and the Hausdorff-Pompeiu excess of subsets, we provide several necessary and/or sufficient conditions of error bounds with mild assumptions. Then we use these primal results to characterize error bounds for composite-convex functions (i.e. the composition of a convex function with a continuously differentiable mapping). It is proved that the primal characterization of error bounds can be established via Bouligand tangent cones, directional derivatives and the Hausdorff-Pompeiu excess if the mapping is metrically regular at the given point. The accurate estimate on the error bound modulus is also obtained.

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arXiv2015

An additive subfamily of enlargements of a maximally monotone operator

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical $ε$-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the $ε$-subdifferential enlargement.

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arXiv2015

Directional Hölder Metric Regularity

This paper sheds new light on regularity of multifunctions through various characterizations of directional Hölder /Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Hölder /Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Hölder /Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed.

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