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

- Cotrina, John, Théra, Michel, Zúñiga, Javier

  • 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

Stability of error bounds for semi-infinite convex constraint systems

- Van Ngai, Huynh, Kruger, Alexander, Théra, Michel

  • 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.

Calmness modulus of linear semi-infinite programs

- Cánovas, Maria, Kruger, Alexander, López, Marco, Parra, Juan, Théra, Michel

  • 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

- Adly, Samir, Hantoute, Abderrahim, Théra, Michel

  • 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.

Perturbation of error bounds

- Kruger, Alexander, López, Marco, Théra, Michel

  • 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.

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

- Huynh, Van Ngai, Théra, Michel

Metric regularity relative to a cone

- Van Ngai, Huynh, Tron, Nguyen, Théra, Michel

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

- Van Ngai, Huynh, Tron, Nguyen, Van Vu, Nguyen, Théra, Michel

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

- Barbagallo, Annamaria, Ernst, Octavian-Emil, Théra, Michel

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

- Gala, Sadek, Ragusa, Maria, Théra, Michel

Set-valued orthogonality and nearness

- Barbagallo, Annamaria, Ernst, Octavian, Théra, Michel

Enlargements of the moreau–rockafellar subdifferential

- Abbasi, Malek, Kruger, Alexander, Théra, Michel

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