Persistency in Discrete Optimization

Abstract

Given a mixed 0–1 linear program in n binary variables, we studied in Chapter 2 the construction of an n + 1 level hierarchy of polyhedral approximations that ranges from the usual continuous relaxation at level 0 to an explicit algebraic representation of the convex hull of feasible solutions at level-n. While exponential in both the number of variables and constraints, level-n was shown in the previous chapter to serve as a useful tool for promoting valid inequalities and facets via a projection operation onto the original variable space. In this chapter, we once again invoke the convex hull representation at level-n, this time to provide a direct linkage between discrete sets and their polyhedral relaxations. Specifically, we study conditions under which a binary variable realizing a value of either 0 or 1 in an optimal solution to the linear programming relaxation of a mixed 0–1 linear program will persist in maintaining that same value in some discrete optimum.