Your selections:

Show More

Show Less

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.

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:

- «
- ‹
- 1
- ›
- »

Are you sure you would like to clear your session, including search history and login status?