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Constructing a perfect duality in infinite programming

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Abstract

One version of the infinite Farkas Lemma states the equivalence of two conditions, (1)y⋅b ⩾ 0 whenevery⋅a j ⩾ 0 forj=1,2,.. and (2)b ∈ cl C, whereb and alla j are inR n andC is the convex cone spanned by all thea j's. In this paper an ascent vector specifies a direction along which an arbitrarily small movement fromb with enterC. A Fredholm type theorem of the alternative characterizes the set of all ascent vectors associated with an arbitrary system of linear inhomogeneous inequalities in a finite number of variables. As a consequence, a pair of infinite programs is constructed which is in perfect duality in the sense that (p1) if one program is consistent and has finite value, then the other is consistent and (p2) if both programs are consistent, then they have the same finite value. The duality is sharp in that the set of all feasible perturbations along rays is determined.

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Authors and Affiliations

  1. Mellon Institute of Science Department of Mathematics, Carnegie-Mellon University, Schenley Park, 15213, Pittsburgh, Pa., USA

    K. O. Kortanek

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  1. K. O. Kortanek
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Additional information

Communicated by J. Stoer

This research was supported by NSF Grants GK-31833 and ENG76-05191. The paper is a revision of an earlier report of June 1975.

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Kortanek, K.O. Constructing a perfect duality in infinite programming. Appl Math Optim 3, 357–372 (1976). https://doi.org/10.1007/BF01448186

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  • DOI: https://doi.org/10.1007/BF01448186

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