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60103 Numerical and Computational Mathematics
2Block perturbations
2Semi-infinite and infinite programming
2Subdifferential
2Variational analysis
10802 Computation Theory and Mathematics
10906 Electrical and Electronic Engineering
11503 Business and Management
1Applications
1Asplund space
1Calmness modulus
1Coderivatives
1Computer science
1Constrained problem
1Convex inequality systems
1Convex infinite inequality systems
1Distance to ill-posedness
1Error bound
1Extremal principle

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Perturbation of error bounds

- Kruger, Alexander, López, Marco, Théra, Michel

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

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

Recent contributions to linear semi-infinite optimization

- Goberna, Miguel, López, Marco

**Authors:**Goberna, Miguel , López, Marco**Date:**2017**Type:**Text , Journal article**Relation:**4OR: A Quarterly Journal of Operations Research Vol. 15, no. 3 (2017), p. 221-264**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented. © 2017, Springer-Verlag GmbH Germany.

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.

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.

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.

Stationarity and regularity of infinite collections of sets

- Kruger, Alexander, López, Marco

**Authors:**Kruger, Alexander , López, Marco**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 154, no. 2 (2012), p. 339-369**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC.

**Authors:**Kruger, Alexander , López, Marco**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 154, no. 2 (2012), p. 339-369**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC.

- 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.

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