A Combinatorial Approach for Small and Strong Formulations of Disjunctive Constraints

Abstract

An important problem in optimization is the construction of mixedinteger programming (MIP) formulations of disjunctive constraints that are both strong and small. Motivated by lower bounds on the number of integer variables that are required by traditional MIP formulations, we present a more general mixed-integer branching formulation framework. Our approach maintains favorable algorithmic properties of traditional MIP formulations: in particular, amenability to branch-and-bound and branch-and-cut algorithms. Our main technical result gives an explicit linear inequality description for both traditional MIP and mixed-integer branching formulations for a wide range of disjunctive constraints. The formulations obtained from this description have linear programming relaxations that are as strong as possible and generalize some of the most computationally effective formulations for piecewise linear functions and other disjunctive constraints. We use this result to produce a strong mixed-integer branching formulation for any disjunctive constraint that uses only two integer variables and a linear number of extra constraints. We sharpen this result for univariate piecewise linear functions and annulus constraints arising in power systems and robotics, producing strong mixed-integer branching formulations that use only two integer variables and a constant (ď 6) number of general inequality constraints. Along the way, we produce two strong logarithmic-sized traditional MIP formulations for the annulus constraint using our main technical result, illustrating its broader utility in the traditional MIP setting.

Ideal, non-extended formulations for disjunctive constraints admitting a network representation

“…Our results complement those of Vielma [45] and Huchette and Vielma [26][27][28]. On the one hand, Vielma [45] was able to derive the facets of the generalized Cayley embedding of SOS2 sets for any choice of the (distinct) binary vectors h i , but our method works for a much wider class of disjunctive sets than SOS2, and we have a full characterization of the non-trivial facets.…”

“…On the one hand, Vielma [45] was able to derive the facets of the generalized Cayley embedding of SOS2 sets for any choice of the (distinct) binary vectors h i , but our method works for a much wider class of disjunctive sets than SOS2, and we have a full characterization of the non-trivial facets. On the other hand, in [26] a linear representation is obtained for the convex hull Q(P, H) of the MIP formulation for combinatorial disjunctive constraints (5) and any choice of distinct binary vectors H = (h i ) m i=1 , while our characterization of facets is valid for any choice of the P i provided that P emb in (6) admits a network representation.…”

“…Tomlin [43] proposed a new modeling of SOS2 sets with binary variables, while Martin et al [38] introduced SOSk constraints and a branching scheme for modeling two-variable piecewise linear functions. Vielma and Nemhauser introduced the concept of independent branching schemes for modeling a constraint very similar to the combinatorial disjunctive constraint (5) of [26], the only difference being that Δ…”

“…such thatx satisfies (1) if and only if (3) admits a feasible solution when x =x (see e.g. [26,32,[44][45][46]). The Linear Programming (LP) relaxation of (3) is the polyhedron Q determined by Ax + By + Cz ≤ b.…”

A geometric way to build strong mixed-integer programming formulations

“…We will focus on the subclass of disjunctive constraints known as combinatorial disjunctive constraints [8]. In particular, we assume that each alternative is a face of the unit simplex…”