List of Titles

Quick View

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
Full Text:
Reviewed:
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

Quick View

Constraint reduction reformulations for projection algorithms with applications to wavelet construction

- Dao, Minh, Dizon, Neil, Hogan, Jeffrey, Tam, Matthew

Authors: Dao, Minh , Dizon, Neil , Hogan, Jeffrey , Tam, Matthew
Date: 2021
Type: Text , Journal article
Relation: Journal of Optimization Theory and Applications Vol. 190, no. 1 (2021), p. 201-233
Full Text:
Reviewed:
Description: We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Quick View

Generalized bregman envelopes and proximity operators

- Burachik, Regina, Dao, Minh, Lindstrom, Scott

Authors: Burachik, Regina , Dao, Minh , Lindstrom, Scott
Date: 2021
Type: Text , Journal article
Relation: Journal of Optimization Theory and Applications Vol. 190, no. 3 (2021), p. 744-778
Full Text:
Reviewed:
Description: Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Quick View

Variational analysis of paraconvex multifunctions

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

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
Full Text:
Reviewed:
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.

Quick View

New tour on the subdifferential of supremum via finite sums and suprema

- Hantoute, A., López-Cerdá, Marco

Authors: Hantoute, A. , López-Cerdá, Marco
Date: 2022
Type: Text , Journal article
Relation: Journal of Optimization Theory and Applications Vol. 193, no. 1-3 (2022), p. 81-106
Full Text:
Reviewed:
Description: This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family. © 2021, The Author(s).

Showing items 1 - 6 of 6