A note on primal-dual stability in infinite linear programming
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A note on primal-dual stability in infinite linear programming Miguel A. Goberna1 · Marco A. López1,2 Virginia N. Vera de Serio4
· Andrea B. Ridolfi3 ·
Received: 12 February 2019 / Accepted: 5 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces. Keywords Linear programming · Infinite dimensions · Primal-dual stability · Consistency · Inconsistency
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Miguel A. Goberna [email protected] Marco A. López [email protected] Andrea B. Ridolfi [email protected] Virginia N. Vera de Serio [email protected]
1
Department of Mathematics, University of Alicante, 03080 Alicante, Spain
2
CIAO, Federation University, Ballarat, Australia
3
Facultad de Ciencias Aplicadas a la Industria, Facultad de Ciencias Económicas, CONICET, Universidad Nacional de Cuyo, Mendoza, Argentina
4
Facultad de Ciencias Económicas, Universidad Nacional de Cuyo, Mendoza, Argentina
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1 Introduction In many practical situations, statisticians, operations researchers or engineers have to minimize a linear functional c, defined on some linear space X (the decision space), subject to a given set of linear constraints at , x ≥ bt , where at is a linear functional on X and bt ∈ R for all t ∈ T . Such a linear problem P:
Inf {c, x : at , x ≥ bt , ∀t ∈ T } ,
x∈X
(1)
called primal, is said to be infinite when the dimension of X and the cardinality of the index set T are both infinite, semi-infinite when X = Rn and T is infinite, and ordinary (or finite) when X = Rn and T is finite. The subdisciplines of optimization dealing with such types of problems are called linear infinite programming (LIP in short), linear semi-infinite programming (LSIP), and (ordinary or finite) linear programming (LP). Classical monographs on the theory, methods and applications of LIP, LSIP and LP are [2,8,13], respectively. The recent literature on duality in LIP has been briefly reviewed in [23, Section 1] and the one of LSIP, in a more detailed way, in [15], which also contains information on LIP. (T ) Let R+ be the positive cone in the space R(
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