A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems

This paper is concerned with the generation of tight equivalent representations for mixedinteger zero-one programming problems. For the linear case, we propose a technique which first converts the problem into a nonlinear, polynomial mixed-integer zero-one problem by multiplying the constraints with some suitable d-degree polynomial factors involving the n binary variables, for any given d E (0, . . . , n}, and subsequently linearizes the resulting problem through appropriate variable transformations. As d varies from zero to n, we obtain a hierarchy of relaxations spanning from the ordinary linear programming relaxation to the convex hull of feasible solutions. The facets of the convex hull of feasible solutions in terms of the original problem variables are available through a standard projection operation. We also suggest an alternate scheme for applying this technique which gives a similar hierarchy of relaxations, but involving fewer “complicating” constraints. Techniques for tightening intermediate level relaxations, and insights and interpretations within a disjunctive programming framework are also presented. The methodology readily extends to multilinear mixed-integer zero-one polynomial programming problems in which the continuous variables appear linearly in the problem.