Gateaux differentiability revisited
- Authors: Abbasi, Malek , Kruger, Alexander , Théra, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Applied Mathematics and Optimization Vol. 84, no. 3 (2021), p. 3499-3516
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
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.
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
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.
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.