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Isolated calmness and sharp minima via Hölder Graphical Derivatives

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.

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

  • Authors: Beer, G. , Cánovas, M. J. , López, Marco , Parra, Juan
  • Date: 2021
  • Type: Text , Journal article
  • Relation: Mathematical Programming Vol. 189, no. 1-2 (2021), p. 75-98. https://purl.org/au-research/grants/arc/DP180100602
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Correction: The article “Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric”, written by Beer,G., Cánovas, M.J., López, M.A., Parra, J.was originally published Online First without Open Access. After publication in volume 189, issue 1–2, page 75–98 the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © The Author(s) 2020 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License. https://doi.org/10.1007/s10107-021-01751-x
  • Description: This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

Necessary and sufficient optimality conditions in DC semi-infinite programming

  • Authors: Correa, Rafael , López, Marco , Pérez-Aros, Pedro
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 837-865
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  • Description: This paper deals with particular families of DC optimization problems involving suprema of convex functions. We show that the specific structure of this type of function allows us to cover a variety of problems in nonconvex programming. Necessary and sufficient optimality conditions for these families of DC optimization problems are established, where some of these structural features are conveniently exploited. More precisely, we derive necessary and sufficient conditions for (global and local) optimality in DC semi-infinite programming and DC cone-constrained optimization, under natural constraint qualifications. Finally, a penalty approach to DC abstract programming problems is developed in the last section. © 2021 Society for Industrial and Applied Mathematics

A note on primal-dual stability in infinite linear programming

  • Authors: Goberna, Miguel , López, Marco , Ridolfi, Andrea , Vera de Serio, Virginia
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Optimization Letters Vol. 14, no. 8 (2020), p. 2247-2263
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  • Description: In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
  • Description: Funding details: Australian Research Council, ARC, DP180100602: http://purl.org/au-research/grants/arc/DP180100602

A uniform approach to hölder calmness of subdifferentials

  • Authors: Beer, Gerald , Cánovas, Maria , López, Marco , Parra, Juan
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Journal of Convex Analysis Vol. 27, no. 1 (2020), p.
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: For finite-valued convex functions f defined on the n-dimensional Euclidean space, we are interested in the set-valued mapping assigning to each pair (f, x) the subdifferential of f at x. Our approach is uniform with respect to f in the sense that it involves pairs of functions close enough to each other, but not necessarily around a nominal function. More precisely, we provide lower and upper estimates, in terms of Hausdorff excesses, of the subdifferential of one of such functions at a nominal point in terms of the subdifferential of nearby functions in a ball centered in such a point. In particular, we obtain the (1/2) - Hölder calmness of our mapping at a nominal pair (f, x) under the assumption that the subdifferential mapping viewed as a set-valued mapping from Rn to Rn with f fixed is calm at each point of {x} × ∂f(x). © Heldermann Verlag
  • Description: Funding details: Australian Research Council, ARC, DP160100854 Funding details: European Commission, EU Funding details: Ministerio de Economía y Competitividad, MINECO Funding details: Federación Española de Enfermedades Raras, FEDER Funding text 1: 

Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems

  • Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
  • Date: 2020
  • Type: Text , Conference paper
  • Relation: Jonathan Borwein Commemorative Conference, JBCC 2017 Vol. 313, p. 43-74
  • Relation: http://purl.org/au-research/grants/arc/DP180100602
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  • Description: We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail. © Springer Nature Switzerland AG 2020.

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 Científico, 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 Científica 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

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.

Calmness of partially perturbed linear systems with an application to the central path

  • Authors: Cánovas, Maria , Hall, Julian , López, Marco , Parra, Juan
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Optimization Vol. 68, no. 2-3 (2019), p. 465-483
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  • Description: In this paper we develop point-based formulas for the calmness modulus of the feasible set mapping in the context of linear inequality systems with a fixed abstract constraint and (partially) perturbed linear constraints. The case of totally perturbed linear systems was previously analyzed in [Canovas MJ, Lopez MA, Parra J, et al. Calmness of the feasible set mapping for linear inequality systems. Set-Valued Var Anal. 2014;22:375-389, Section 5]. We point out that the presence of such an abstract constraint yields the current paper to appeal to a notable different methodology with respect to previous works on the calmness modulus in linear programming. The interest of this model comes from the fact that partially perturbed systems naturally appear in many applications. As an illustration, the paper includes an example related to the classical central path construction. In this example we consider a certain feasible set mapping whose calmness modulus provides a measure of the convergence of the central path. Finally, we underline the fact that the expression for the calmness modulus obtained in this paper is (conceptually) implementable as far as it only involves the nominal data.

Convexity and closedness in stable robust duality

  • Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Optimization Letters Vol. 13, no. 2 (2019), p. 325-339
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.

Best approximate solutions of inconsistent linear inequality systems

Indexation strategies and calmness constants for uncertain linear inequality systems

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.

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.

A unifying approach to robust convex infinite optimization duality

Farkas-type results for vector-valued functions with applications

  • Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.
  • Date: 2017
  • Type: Text , Journal article
  • Relation: Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390
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  • Description: The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

Recent contributions to linear semi-infinite optimization

Towards supremum-sum subdifferential calculus free of qualification conditions

  • Authors: Correa, Rafael , Hantoute, Abderrahim , López, Marco
  • Date: 2016
  • Type: Text , Journal article
  • Relation: Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

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
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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.