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11Goberna, Miguel
8Cánovas, Maria
8Parra, Juan
5Correa, Rafael
4Dinh, Nguyen
4Hantoute, Abderrahim
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3Vera De Serio, Virginia
2Mo, T. H.
2Ridolfi, Andrea
2Rubinov, Alex
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Comparative study of RPSALG algorithm for convex semi-infinite programming

- Auslender, Alfred, Ferrer, Albert, Goberna, Miguel, López, Marco

**Authors:**Auslender, Alfred , Ferrer, Albert , Goberna, Miguel , López, Marco**Date:**2014**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 60, no. 1 (2014), p. 59-87**Full Text:**false**Reviewed:****Description:**The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

A uniform approach to hölder calmness of subdifferentials

- Beer, Gerald, Cánovas, Maria, López, Marco, Parra, Juan

**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**Full Text:**false**Reviewed:****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:

Weaker conditions for subdifferential calculus of convex functions

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**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**Reviewed:****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.

Towards supremum-sum subdifferential calculus free of qualification conditions

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

Subdifferential of the supremum via compactification of the index set

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**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**Full Text:****Reviewed:****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, TecnolÃ³gico y de InnovaciÃ³n TecnolÃ³gica, 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: ComisiÃ³n Nacional de InvestigaciÃ³n CientÃfica y TecnolÃ³gica, CONICYT Funding details: Ministerio de Ciencia e InnovaciÃ³n, 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

**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**Full Text:****Reviewed:****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, TecnolÃ³gico y de InnovaciÃ³n TecnolÃ³gica, 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: ComisiÃ³n Nacional de InvestigaciÃ³n CientÃfica y TecnolÃ³gica, CONICYT Funding details: Ministerio de Ciencia e InnovaciÃ³n, 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

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**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**Full Text:**false**Reviewed:****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.

Necessary and sufficient optimality conditions in DC semi-infinite programming

- Correa, Rafael, López, Marco, Pérez-Aros, Pedro

**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**Full Text:****Reviewed:****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

**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**Full Text:****Reviewed:****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

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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

Calmness of the feasible set mapping for linear inequality systems

- Cánovas, Maria, López, Marco, Parra, Juan, Toledo, Javier

**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**Reviewed:****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.

- Cánovas, Maria, López, Marco, Parra, Juan, Toledoa, Javier

**Authors:**Cánovas, Maria , López, Marco , Parra, Juan , Toledoa, Javier**Date:**2011**Type:**Text , Journal article**Relation:**Optimization Vol. 60, no. 7 (2011), p. 925-946**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**This article extends some results of Cá novas et al. [M.J. Cá novas, M.A. Ló pez, J. Parra, and F.J. Toledo, Distance to ill-posedness and the consistency value of linear semi-infinite inequality systems, Math. Prog. Ser. A 103 (2005), pp. 95-126.] about distance to ill-posedness (feasibility/ infeasibility) in three directions: From individual perturbations of inequalities to perturbations by blocks, from linear to convex inequalities and from finite- to infinite-dimensional (Banach) spaces of variables. The second of the referred directions, developed in the finite-dimensional case, was the original motivation of this article. In fact, after linearizing a convex system via the Fenchel-Legendre conjugate, affine perturbations of convex inequalities translate into block perturbations of the corresponding linearized system. We discuss the key role played by constant perturbations as an extreme case of block perturbations. We emphasize the fact that constant perturbations are enough to compute the distance to ill-posedness in the infinite-dimensional setting, as shown in the last part of this article, where some remarkable differences of infinite- versus finite-dimensional systems are presented. Throughout this article, the set indexing the constraints is arbitrary, with no topological structure. Accordingly, the functional dependence of the system coefficients on the index has no qualification at all.

Indexation strategies and calmness constants for uncertain linear inequality systems

- Cánovas, Maria, Henrion, René, López, Marco, Parra, Juan

**Authors:**Cánovas, Maria , Henrion, René , López, Marco , Parra, Juan**Date:**2018**Type:**Text , Book chapter**Relation:**The Mathematics of the Uncertain (part of the Studies in Systems, Decision and Control series) p. 831-843**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**The present paper deals with uncertain linear inequality systems viewed as nonempty closed coefficient sets in the (n+ 1) -dimensional Euclidean space. The perturbation size of these uncertainty sets is measured by the (extended) Hausdorff distance. We focus on calmness constants—and their associated neighborhoods—for the feasible set mapping at a given point of its graph. To this aim, the paper introduces an appropriate indexation function which allows us to provide our aimed calmness constants through their counterparts in the setting of linear inequality systems with a fixed index set, where a wide background exists in the literature.

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

- Cánovas, Maria, López, Marco, Mordukhovich, Borris, Parra, Juan

**Authors:**Cánovas, Maria , López, Marco , Mordukhovich, Borris , Parra, Juan**Date:**2012**Type:**Text , Journal article**Relation:**TOP Vol. 20, no. 2 (2012), p. 310-327**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504-1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

**Authors:**Cánovas, Maria , López, Marco , Mordukhovich, Borris , Parra, Juan**Date:**2012**Type:**Text , Journal article**Relation:**TOP Vol. 20, no. 2 (2012), p. 310-327**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504-1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

Subdifferentials and stability analysis of feasible set and pareto front mappings in linear multiobjective optimization

- Cánovas, Maria, López, Marco, Mordukhovich, Boris, Parra, Juan

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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

- Cánovas, Maria, Hall, Julian, López, Marco, Parra, Juan

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

Lower semicontinuity of the feasible set mapping of linear systems relative to their domains

- Daniilidis, Aris, Goberna, Miguel, López, Marco, Lucchetti, Roberto

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

From the Farkas lemma to the Hahn-Banach theorem

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Mo, T. H.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 2 (2014), p. 678-701**Full Text:****Reviewed:****Description:**This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) â‰¥ 0 which are consequences of a composite convex inequality (S Â° g)(x) â‰¤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 2 (2014), p. 678-701**Full Text:****Reviewed:****Description:**This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) â‰¥ 0 which are consequences of a composite convex inequality (S Â° g)(x) â‰¤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.

A unifying approach to robust convex infinite optimization duality

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.

Farkas-type results for vector-valued functions with applications

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Mo, T. H.

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

Convexity and closedness in stable robust duality

- Dinh, Nguyen, Goberna, Miguel, López, Marco, Volle, Michel

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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

- Goberna, Miguel, Hiriart-Urruty, Jean-Baptiste, López, Marco

**Authors:**Goberna, Miguel , Hiriart-Urruty, Jean-Baptiste , López, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 271-284**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper is intended to characterize three types of best approximate solutions for inconsistent linear inequality systems with an arbitrary number of constraints. It also gives conditions guaranteeing the existence of best uniform solutions and discusses potential applications.

**Authors:**Goberna, Miguel , Hiriart-Urruty, Jean-Baptiste , López, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 271-284**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper is intended to characterize three types of best approximate solutions for inconsistent linear inequality systems with an arbitrary number of constraints. It also gives conditions guaranteeing the existence of best uniform solutions and discusses potential applications.

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