Generalized cross decomposition applied to nonlinear integer programming problems: duality gaps and convexification in parts

Abstract

In this paper we study the lower bounds on the optimal objective function value of nonlinear pure integer programming problems obtainable by convexification in parts, achieved by using generalized Benders or cross decomposition, and compare them to the best lower bounds obtainable by the convexification introduced by the Lagrangean dual, i.e. by Lagrangean relaxation together with subgradient optimization or (nonlinear) Dantzig-Wolfe decomposition. We show how to obtain a number of different bounds and specify the known relations between them. In one case generalized cross decomposition can automatically yield the best of the Benders decomposition bound and the Lagrangean dual bound, without any a priori knowledge of which is the best. In another case the cross decomposition bound dominates the Lagrangean dual bound.