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New glimpses on convex infinite optimization duality

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

  • Authors: Goberna, Miguel , López, Marco , Volle, Michel
  • Date: 2015
  • Type: Text , Journal article
  • Relation: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas Vol. 109, no. 2 (2015), p. 431-450
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  • Description: Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that (P)=max(D), (P)=max(Q), and (P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing (P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described. © 2014, Springer-Verlag Italia.

Recent contributions to linear semi-infinite optimization

- Goberna, Miguel, López, Marco

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
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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.

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
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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.

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
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  • 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.

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
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  • 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.

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

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

Distance to ill-posedness for linear inequality systems under block perturbations : Convex and infinite-dimensional cases

- 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
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  • 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

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
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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.

On coderivatives and Lipschitzian properties of the dual pair in optimization

- López, Marco, Ridolfi, Andrea, Vera De Serio, Virginia

  • Authors: López, Marco , Ridolfi, Andrea , Vera De Serio, Virginia
  • Date: 2012
  • Type: Text , Journal article
  • Relation: Nonlinear Analysis, Theory, Methods and Applications Vol. 75, no. 3 (2012), p. 1461-1482
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  • Description: In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem. © 2011 Elsevier Ltd. All rights reserved.

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
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  • 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.

Recent contributions to linear semi-infinite optimization : An update

- Goberna, Miguel, López, Marco

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
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  • Description: This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ° g)(x) ≤ 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.
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