Disjunctive cuts for cross-sections of the second-order cone

Sercan Yıldız, Gérard Cornuéjols
2015 Operations Research Letters  
In this paper we study general two-term disjunctions on affine cross-sections of the secondorder cone. Under some mild assumptions, we derive a closed-form expression for a convex inequality that is valid for such a disjunctive set, and we show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions on ellipsoids and paraboloids and a wide class of two-term disjunctions-including split disjunctions-on hyperboloids. Our approach relies on the work
more » ... Kılınç-Karzan and Yıldız which considers general two-term disjunctions on the second-order cone. by exploiting the integrality of the variables x j , j ∈ J, and enhancing C with linear two-term disjunctions l 1 x ≥ l 1,0 ∨ l 2 x ≥ l 2,0 that are satisfied by all solutions in S. Valid inequalities that are obtained from disjunctions using this approach are known as disjunctive cuts. In this paper we study two-term disjunctions on the set C and give closed-form expressions for the strongest disjunctive cuts that can be obtained from such disjunctions. Disjunctive cuts were introduced by Balas in the context of mixed-integer linear programming [3] and have since been the cornerstone of theoretical and practical achievements in integer programming. There has been a lot of recent interest in extending disjunctive cutting-plane theory from the domain of mixed-integer linear programming to that of mixed-integer conic programming [2, 7, 9, 11, 12, 18] . Kılınç-Karzan [13] studied minimal valid linear inequalities for general disjunctive conic sets and showed that these are sufficient to describe the associated closed convex hull under a mild technical assumption. Bienstock and Michalka [6] studied the characterization and separation of linear inequalities that are valid for the epigraph of a convex, differentiable function whose domain is restricted to the complement of a convex set. On the other hand, several papers in the last few years have focused on deriving closed-form expressions for nonlinear convex inequalities that fully describe the convex hull of a disjunctive second-order conic set in the space of the original variables. Dadush et al. [10] and Andersen and Jensen [1] derived split cuts for ellipsoids and the second-order cone, respectively. Modaresi et al. extended these results to split disjunctions on cross-sections of the second-order cone [16] and compared the effectiveness of split cuts against conic MIR inequalities and extended formulations [15] . For disjoint two-term disjunctions on cross-sections of the second-order cone and under the assumption that {x ∈ C : l 1 x = l 1,0 } and {x ∈ C : l 2 x = l 2,0 } are bounded, Belotti et al. [4, 5] proved that there exists a unique cone which describes the convex hull of the disjunction. They also identified a procedure for identifying this cone when C is an ellipsoid. Using the structure of minimal valid linear inequalities, Kılınç-Karzan and Yıldız [14] derived a family of convex inequalities which describes the convex hull of a general two-term disjunction on the whole second-order cone. In this paper, we pursue a similar goal: We study general two-term disjunctions on a cross-section C of the second-order cone, namely C = {x ∈ L n : Ax = b}.
doi:10.1016/j.orl.2015.06.001 fatcat:5jzzbgaofvbwzpiwzf54jqtqb4