Optimization Online


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.


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

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Programming Society