On the Stability of the Extreme Point Set in Linear Optimization

Goberna, Miguel Á.; Larriqueta, Mercedes; Serio, Virginia Verade

doi:10.1137/040607927

Cited by 18 publications

(15 citation statements)

References 12 publications

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“…We are concerned with those parameters which are stable for the corresponding property in the sense that sufficiently small perturbations of the parameter preserve its membership. The interior of Π P C , Π P IC , Π D C , and Π D IC where characterized in [9] and [10], whereas [3], [12] and [13] have characterized the interior of the elements of the primal partition (of Π), Π P IC , Π P B , Π P U B , the dual partition,…”

Section: Introductionmentioning

confidence: 99%

Generic primal-dual solvability in continuous linear semi-infinite programming

Goberna

Todorov²

2008

Optimization

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In this paper we consider the space of all the linear semi-infinite programming (LSIP) problems with a given infinite compact Hausdorff index set, a given number of variables, and continuous coefficients, endowed with the topology of the uniform convergence. These problems are classified as inconsistent, solvable with bounded optimal set, bounded (i.e., finite valued) but either unsolvable or having an unbounded optimal set, and unbounded (i.e., with infinite optimal value), giving rise to the socalled refined primal partition of the space of problems. The mentioned LSIP problems can be also classified with a similar criterion applied to the corresponding Haar's dual problems, which provides the refined dual partition of the space of problems. We characterize the interior of the elements of the refined primal and dual partitions as well as the interior of the intersections of the elements of both partitions (the so-called refined primal-dual partition). These characterizations allow to prove that most (primal or dual) bounded problems have simultaneously primal and dual non-empty bounded optimal set. Consequently, most bounded continuous LSIP problems are primal and dual solvable.

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Section: Introductionmentioning

confidence: 99%

Generic primal-dual solvability in continuous linear semi-infinite programming

Goberna

Todorov²

2008

Optimization

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“…In the same fashion, is LHS-upper bounded if sup fka t k ; t 2 T g < +1 (with an analogous interpretation, just replacing the origin by the point at in…nity). Moreover, we say that is absolutely stable if its feasible set is nonempty and remains constant under arbitrary but su¢ ciently small perturbations of all the data (a; b; c) : In Example 1 of [8], it is shown, for any in…nite index set T , that admits a reformulation in (with the same objective function, and feasible set assumed to be bounded) which is absolutely stable.…”

Section: Preliminariesmentioning

confidence: 99%

On stable uniqueness in linear semi-infinite optimization

2011

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Abstract. This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-in…nite optimization (LSIO) problem, in the sense of the maintaining of the strong uniqueness property under su¢ ciently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (a¢ nely) stable problems, i.e., those LSIO problems whose feasible set (its a¢ ne hull, respectively) remains constant under su¢ ciently small perturbations.

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“…Thus, int ) can be seen as the set of primal (dual) stable problems. These interiors have been characterized in [8], [7] and [9]. On the other hand, [12] de…nes a conic programming problem to be ill-posed (in primal-dual feasibility sense) when it lays on the boundary of the set of consistent problems whose corresponding dual is also consistent.…”

Section: Introductionmentioning

confidence: 99%

Primal, dual and primal-dual partitions in continuous linear optimization

Goberna

Todorov

2007

Optimization

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We associate with each natural number n and each compact Hausdor¤ topological space T the space of linear optimization problems with n primal variables and index set T (for the constraints) equipped with the topology of the uniform convergence. We consider three di¤erent partitions of this pseudometric space. The primal and the dual partitions are the result of classifying a given optimization problem and its dual as either inconsistent or bounded or unbounded, whereas the primal-dual partition is formed by the non-empty intersections of the elements of both partitions. The elements of the three partitions are neither open nor closed and their topological interiors are formed by those problems for which su¢ -ciently small perturbations maintain the membership of the problem, i..e., the problems that are stable for the corresponding property. We prove that the stable problems are the same for the three partitions, concluding that most problems are stable in the three senses. This is done by completing the topological analysis of the primal-dual partition carried out in a previous paper of the authors.

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On the Stability of the Extreme Point Set in Linear Optimization

Cited by 18 publications

References 12 publications

Generic primal-dual solvability in continuous linear semi-infinite programming

Generic primal-dual solvability in continuous linear semi-infinite programming

On stable uniqueness in linear semi-infinite optimization

Primal, dual and primal-dual partitions in continuous linear optimization

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