On the sufficiency of finite support duals in semi-infinite linear programming

Finite-dimensional linear programs satisfy strong duality (SD) and have the "dual pricing" (DP) property. The (DP) property ensures that, given a sufficiently small perturbation of the right-hand-side vector, there exists a dual solution that correctly "prices" the perturbation by computing the exact change in the optimal objective function value. These properties may fail in semi-infinite linear programming where the constraint vector space is infinite dimensional. Unlike the finite-dimensional case, in semi-infinite linear programs the constraint vector space is a modeling choice. We show that, for a sufficiently restricted vector space, both (SD) and (DP) always hold, at the cost of restricting the perturbations to that space. The main goal of the paper is to extend this restricted space to the largest possible constraint space where (SD) and (DP) hold. Once (SD) or (DP) fail for a given constraint space, then these conditions fail for all larger constraint spaces. We give sufficient conditions for when (SD) and (DP) hold in an extended constraint space. Our results require the use of linear functionals that are singular or purely finitely additive and thus not representable as finite support vectors. The key to understanding these linear functionals is the extension of the Fourier-Motzkin elimination procedure to semi-infinite linear programs.

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“…where the inequality was shown in [3]. Also, for a fixed δ ≥ 0, sup i∈N 2 i − 1 i 2 δ ≥ 0 and so ω(δ, b) ≥ 0 for all δ ≥ 0.…”

Section: Dual Pricing In Extended Constraint Spacesmentioning

confidence: 89%

“…Thus, the hypotheses of Lemma A.2 hold. Now apply Lemma A.2 and observe there is aˆ which we can take to be less than 3 z − x 1 ≥ 0…”

Section: Dual Pricing In Extended Constraint Spacesmentioning

confidence: 99%

Strong duality and sensitivity analysis in semi-infinite linear programming

2016

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“…To the best of our knowledge, the …rst semi-in…nite version of that method, which provides linear representations of the projections of closed convex sets on the coordinate hyperplanes, was introduced in [77] to characterize the socalled Motzkin decomposable sets (i.e., those sets which can be expressed as sums of polyhedral convex sets with closed convex cones, as the optimal set S of P when c 2 rint M ), see also [81] and [77]. The second and third semi-in…nite versions of the Fourier elimination method are due to A. Basu, K. Martin, and C. Ryan ( [12], [13], [14]) and to K. Kortanek and Q. Zhang [127], respectively, these four papers dealing with LSIO duality theory.…”

Section: Dualitymentioning

confidence: 99%

“…A duality scheme for LSIO inspired by [7] is used in [12] (where the index set T is countable), [13], and [14]. Denoting by Y a linear subspace of R T (called constraint space) such that U := span fa 1 ( ) ; :::; a n ( ) ; b ( )g Y;…”

Section: Dualitymentioning

confidence: 99%

Recent contributions to linear semi-infinite optimization

Goberna

López

2017

4OR-Q J Oper Res

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This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-in…nite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of …elds. Extensions to related optimization areas, as convex semi-in…nite optimization, linear in…nite optimization, and multi-objective linear semi-in…nite optimization, are also commented.

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“…Fortunately, when the constraint space vector space is R N , all positive dual functionals are countably additive. Indeed, Basu, Martin and Ryan [9] prove that positive dual functionals in the algebraic dual of R N can be expressed as positive sequences with finite support; that is, ψ i > 0 for only finitely many i ∈ N. Clearly such dual functionals are countably additive. Then the resulting dual program is derived by taking the transpose of the constraint matrix in (1.6):…”

Section: Motivationmentioning

confidence: 99%

The Slater Conundrum: Duality and Pricing in Infinite-Dimensional Optimization

Martin¹,

Ryan²,

Stern³

2016

SIAM J. Optim.

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Duality theory is pervasive in finite dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a vector of "dual prices" that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are σ-order complete Riesz spaces (ordered vector spaces with a lattice structure). In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slater's condition) imply the existence of singular dual solutions that are difficult to find and interpret. We call this phenomenon the Slater conundrum: interior points ensure zero duality gap (a desirable property), but interior points also imply the existence of singular dual solutions (an undesirable property). A Riesz space is the most parsimonious vector-space structure sufficient to characterize the Slater conundrum. Topological vector spaces common to the vast majority of infinite dimensional optimization literature are not necessary.

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On the sufficiency of finite support duals in semi-infinite linear programming

Cited by 9 publications

References 13 publications

Strong duality and sensitivity analysis in semi-infinite linear programming

Strong duality and sensitivity analysis in semi-infinite linear programming

Recent contributions to linear semi-infinite optimization

The Slater Conundrum: Duality and Pricing in Infinite-Dimensional Optimization

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