Two-Term Disjunctions on the Second-Order Cone

Abstract

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.

On sublinear inequalities for mixed integer conic programs

“…In particular, classes of minimal and sublinear inequalities, where these notions are defined with respect to the cone K, had received quite some interest in the recent years. For example, such a study on the structure of tight, K-minimal inequalities underlies the developments of nonlinear disjunctive cuts in [23,24] for the case of two-term linear disjunctions applied to a second-order cone.…”

“…For example, such sets cover the simpler setups commonly studied such as the two-term disjunctions or split disjunctions on regular cones or their cross-sections. The particular case of two-term or split disjunctions on K = L n has recently attracted a lot of attention [1,2,6,7,9,12,15,[23][24][25][26]30].…”

Convex hull of two quadratic or a conic quadratic and a quadratic inequality

“…⊓ ⊔ Sets considered by Corollaries 1-3 that are unions of convex sets include those constructed from two-term disjunctions such as the ones considered in [15,Section 6] and [2,3,4,5,6,7,9,10,13,16,21,22,23,25,30,32,36,37,46]. Such sets are the unions of two convex sets defined by a single quadratic or conic quadratic inequality and two linear inequalities.…”

“…In particular, the works in [3,4,16,21,22,23] study cuts from one-sided split disjunctions, and such cuts cannot be derived using Lemma 4. On the other hand, the works in [5,6,7,32,46] assume the non-emptiness of the interiors of both sides of the disjunctions in the derivation of the cuts, and such cuts can in fact be derived using Lemma 4. Finally, it should be noted that the assumption from [15,Section 6] that seems of relevance to Lemma 4, i.e., [15,Assumption 2], only assumes that a single set A i needs to have nonempty interior.…”

“…Morán et al [40] consider subadditive inequalities for general Mixed Integer Conic Programming and Kılınç-Karzan [31] studies minimal valid linear inequalities to characterize the convex hull of general conic sets with a disjunctive structure. Following the results in [31], Kılınç-Karzan and Yıldız [32] study the structure of the convex hull of a two-term disjunction applied to the second-order cone. Yıldız and Cornuéjols [46] study disjunctive cuts on cross sections of the second-order cone.…”