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Generating functions and duality for integer programs

Jean B. Lasserre (lasserre***at***laas.fr)

Abstract: We consider the integer program P -> max {c'x | Ax=y; x\in N^n }. Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables $\lambda\in R^m$ have discrete analogues $z\in C^m$, related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P*, an analogue of the dual LP in linear programming.

Keywords: generating functions; integer programming; duality

Category 1: Integer Programming

Category 2: Global Optimization (Theory )

Citation: Discrete Optimization 1 (2004), pp. 167--187.

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Entry Submitted: 12/02/2003
Entry Accepted: 12/02/2003
Entry Last Modified: 09/23/2005

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