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Duality and a Farkas lemma for integer programs

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Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 32))

Abstract

We consider the integer program \(\max \{c x\,|\,Ax=b,x \in {\bf N}^n\}\). A formal parallel between linear programming and continuous integration, and discrete summation, shows that a natural duality for integer programs can be derived from the \({\bf Z}\)-transform and Brion and Vergne’s counting formula. Along the same lines, we also provide a discrete Farkas lemma and show that the existence of a nonnegative integral solution \(x\in{\bf N}^n\) to \(Ax=b\) can be tested via a linear program.

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  1. LAAS-CNRS, 31077, 7 Avenue du Colonel Roche, Toulouse Cédex 4, France

    Jean B. Lasserre

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  1. Jean B. Lasserre
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Correspondence to Jean B. Lasserre .

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    © 2009 Springer-Verlag New York

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    Lasserre, J.B. (2009). Duality and a Farkas lemma for integer programs. In: Pearce, C., Hunt, E. (eds) Optimization. Springer Optimization and Its Applications, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-0-387-98096-6_2

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