Intersection cuts for nonlinear integer programming: convexification techniques for structured sets

Abstract

We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single quadratic inequality. We also give simple formulas for k-branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets.

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

“…However, the construction of these cuts does not exploit the structure induced by the section and they hence cannot be expected to always achieve the full strength of the nonlinear split cuts for E studied in [4,14,21]. The following proposition shows that this is indeed the case and that the cut with the weakest effect on E is the CMIR.…”

“…However, for special classes of K, we can characterize the nonlinear split cuts that need to be added to K to obtain K π,π 0 [1,4,5,14,19,21]. For instance, the following proposition from [21] characterizes split cuts for conic quadratic sets of the form…”

“…Finally, we show that the containment is strict for n = 2, B = I, c = (1/4, 0) T , π = (1, 1) T , and π 0 = 0. Again using Corollary 5 in [21], one can check that after a few simplifications, the corresponding split cut is…”

“…The CMIR and nonlinear split cuts for C and Q (characterized in [21]) can be used to induce valid inequalities for E ∩ Z n through the same linear section of Lemma 2. However, the construction of these cuts does not exploit the structure induced by the section and they hence cannot be expected to always achieve the full strength of the nonlinear split cuts for E studied in [4,14,21].…”

“…The study of split cuts for MINLP is still much more limited than for MILP; however, there has been significant work on the computational use of split cuts in MINLP [9,11,16,18,25] and a recent surge of theoretical developments [1,2,4,5,14,19,21,22]. In particular, several formulas for split cuts for Mixed Integer Conic Quadratic Programming (MICQP) have been recently developed [1,4,5,14,21]. While the resulting cuts are strong nonlinear inequalities, adding these cuts to the continuous relaxation of the MICQP can significantly increase its solution time, which could negate the effectiveness of the cuts.…”