A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

Abstract

We study a generic minimization problem with separable non-convex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.Key words: piecewise-linear, integer programming, linear relaxation, Lagrangian relaxation. Resum6Nous considerons un probleme de minimisation gen6rique dans lequel l'objectif consiste d'une somme separable de fonctions lineaires par morceaux non convexes. Nous montrons que les relaxations lineaires de trois modeles classiques de programmation en nombres entiers sont equivalentes puisqu'elles fournissent comme approximation de l'objectif son enveloppe convexe inf6rieure. Nous etablissons galement une relation entre ce rsultat et la th6orie de la dualite lagrangienne classique.Mots-cls : lineaire par morceaux, progrmmation en nombres entiers, relaxation lin6aire, relaxation lagrangienne.ii

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