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Characterizations of stability of error bounds for convex inequality constraint systems

  • Authors: Wei, Zhou , Théra, Michel , Yao, Jen-Chih
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Open Journal of Mathematical Optimization Vol. 3, no. (2022), p. 1-17
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  • Description: In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimum of the directional derivative at a reference point over the unit sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the unit sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving convex minimization problems (defined by directional derivatives) over the unit sphere. © Zhou Wei & Michel Théra & Jen-Chih Yao.

Franco Giannessi : “Every mathematician can do a true theorem : only a genius can make an important mistake”

Variational analysis of paraconvex multifunctions

  • Authors: Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 180-218
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: Our aim in this article is to study the class of so-called ρ- paraconvex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function ρ: X→ [0 , + ∞) satisfying some suitable conditions. This class of multifunctions generalizes the class of γ- paraconvex multifunctions with γ> 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

Enlargements of the moreau–rockafellar subdifferential

Gateaux differentiability revisited

Logarithmically improved regularity criterion for the 3D Hall-MHD equations

  • Authors: Gala, Sadek , Théra, Michel
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Computational and Applied Mathematics Vol. 40, no. 7 (2021), p.
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  • Description: In this work, we study the blow-up criterion of the smooth solutions of three-dimensional incompressible Hall-magnetohydrohynamics equations (in short, Hall-MHD). We obtain a logarithmically improved regularity criterion of smooth solutions in terms of the B˙∞,∞0 norm. We improve the blow-up criterion for smooth solutions established in Ye (Appl Anal 96:2669–2683, 2016). © 2021, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

On the regularity of weak solutions of the boussinesq equations in besov spaces

Strongly regular points of mappings

  • Authors: Abbasi, Malek , Théra, Michel
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (Journal article 2021), p.
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  • Description: In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound. © 2021, The Author(s).

An existence result for quasi-equilibrium problems via Ekeland’s variational principle

  • Authors: Cotrina, John , Théra, Michel , Zúñiga, Javier
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 187, no. 2 (2020), p. 336-355
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  • Description: This paper deals with the existence of solutions to equilibrium and quasi-equilibrium problems without any convexity assumption. Coverage includes some equivalences to the Ekeland variational principle for bifunctions and basic facts about transfer lower continuity. An application is given to systems of quasi-equilibrium problems. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
  • Description: Research of M. Théra is supported by the Australian Research Council (ARC) Grant DP160100854 and benefited from the support of the FMJH Program PGMO and from the support of EDF. http://purl.org/au-research/grants/arc/DP160100854

Directional metric pseudo subregularity of set-valued mappings: a general model

Orthogonality in locally convex spaces : two nonlinear generalizations of Neumann's lemma

Set-valued orthogonality and nearness

Metric regularity relative to a cone

Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions

Perturbation of error bounds

  • Authors: Kruger, Alexander , López, Marco , Théra, Michel
  • Date: 2018
  • Type: Text , Journal article
  • Relation: Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Calmness modulus of linear semi-infinite programs

  • Authors: Cánovas, Maria , Kruger, Alexander , López, Marco , Parra, Juan , Théra, Michel
  • Date: 2014
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 29-48
  • Relation: http://purl.org/au-research/grants/arc/DP110102011
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  • Description: Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions

  • Authors: Adly, Samir , Hantoute, Abderrahim , Théra, Michel
  • Date: 2012
  • Type: Text , Journal article
  • Relation: Nonlinear Analysis: Theory, Methods & Applications Vol. 75, no. 3 (February, 2012), p. 985-1008
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  • Description: The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.

Stability of error bounds for semi-infinite convex constraint systems

  • Authors: Van Ngai, Huynh , Kruger, Alexander , Théra, Michel
  • Date: 2010
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 20, no. 4 (2010), p. 2080-2096
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  • Description: In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its "small" perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error bounds for semi-infinite systems of convex inequalities. By applying these characterizations, we extend some results established by Azé and Corvellec [SIAM J. Optim., 12 (2002), pp. 913-927] on the sensitivity analysis of Hoffman constants to semi-infinite linear constraint systems. Copyright © 2010, Society for Industrial and Applied Mathematics.
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