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.KeywordsDual SolutionContinuous RelaxationComplementary SlacknessOptimal Dual SolutionPersistency PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.