Calmness of the feasible set mapping for linear inequality systems
- Authors: Cánovas, Maria , López, Marco , Parra, Juan , Toledo, Javier
- Date: 2014
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 22, no. 2 (2014), p. 375-389
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system's data.
Lower semicontinuity of the feasible set mapping of linear systems relative to their domains
- Authors: Daniilidis, Aris , Goberna, Miguel , López, Marco , Lucchetti, Roberto
- Date: 2013
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 21, no. 1 (2013), p. 67-92
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.
Weaker conditions for subdifferential calculus of convex functions
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Functional Analysis Vol. 271, no. 5 (2016), p. 1177-1212
- Relation: http://purl.org/au-research/grants/arc/DP160100854
- Full Text: false
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- Description: In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization.
- Description: In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization. (C) 2016 Elsevier Inc. All rights reserved.
Subdifferential of the closed convex hull of a function and integration with nonconvex data in general normed spaces
- Authors: López, Marco , Volle, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 390, no. 1 (2012), p. 307-312
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: In this paper we approach the study of the subdifferential of the closed convex hull of a function and the related integration problem without the usual assumption of epi-pointedness. The key tool is, as in Hiriart-Urruty et al. (2011) [7], the concept of ε-subdifferential. Some other assumptions which are standard in the literature are also removed.
Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus
- Authors: Correa, R. , Hantoute, A. , López, Marco
- Date: 2021
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.
Subdifferential of the supremum via compactification of the index set
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2020
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 48, no. 3 (2020), p. 569-588, http://purl.org/au-research/grants/arc/DP180100602
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- Description: We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
- Description: Funding details: Fondo Nacional de Desarrollo CientÃfico, Tecnológico y de Innovación Tecnológica, FONDECYT, PIA AFB-170001, 1190110, 1190012 Funding details: Universidad de Alicante, BEA- GAL 18/00205, PGC2018-097960-B-C21 Funding details: Australian Research Council, ARC, DP 180100602 Funding details: Comisión Nacional de Investigación CientÃfica y Tecnológica, CONICYT Funding details: Ministerio de Ciencia e Innovación, MICINN Funding text 1: Research supported by CONICYT (Fondecyt 1190012 and 1190110), Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Grant Beatriz Galindo BEA- GAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC - Discovery Projects DP 180100602
Valadier-like formulas for the supremum function I
- Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 25, no. 4 (2018), p. 1253-1278
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We generalize and improve the original characterization given by Valadier [19, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given in [11, Theorem 4], which uses the e-subdifferential at the reference point.
Subdifferentials and stability analysis of feasible set and pareto front mappings in linear multiobjective optimization
- Authors: Cánovas, Maria , López, Marco , Mordukhovich, Boris , Parra, Juan
- Date: 2020
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 315-334
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: The paper concerns multiobjective linear optimization problems in
- Description: Funding details: European Commission, EC Funding details: European Regional Development Fund, FEDER Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP180100602 Funding details: Australian Research Council, ARC, DP-190100555 Funding details: Air Force Office of Scientific Research, AFOSR, 15RT04 Funding details: DMS-1512846, DMS-1808978 Funding text 1: This research has been partially supported by grants MTM2014-59179-C2-(1,2)-P and PGC2018-097960-B-C2(1,2) from MINECO/MICINN, Spain, and ERDF, “A way to make Europe”, European Union. Funding text 2: Research of the second author is also partially supported by the Australian Research Council (ARC) Discovery Grants Scheme (Project Grant # DP180100602). Funding text 3: Research of third author was partially supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research grant #15RT04, and by Australian Research Council under grant DP-190100555.