Nonlinear 0–1 programming: II. Dominance relations and algorithms

Abstract

A nonlinear 0–1 program can be restated as a multilinear 0–1 program, which in turn is known to be equivalent to a linear 0–1 program with generalized covering (g.c.) inequalities. In a companion paper [6] we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0–1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time to an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12–15, and the difference in their performance grows with the number of such terms.