n-step Conic Mixed Integer Rounding Inequalities

We introduce the n-step conic MIR inequalities for the so-called polyhedral second-order conic (PSOC) mixed integer sets. PSOC sets arise in the polyhedral reformulation of the second-order conic mixed integer programs. Moreover, they are an equivalent representation for any mixed integer set defined by two linear constraints. The simple conic MIR inequalities of Atamt├╝rk and Narayanan (Math Program 122:1-20, 2010) and the n-step MIR inequalities of Kianfar and Fathi (Math Program 120:313-346, 2009) are special cases of the n-step conic MIR inequalities. We first derive the n-step conic MIR inequality for a PSOC set with n integer variables and prove that all the 1-step to n-step conic MIR inequalities are facet-defining for the convex hull of this set. We also provide necessary and sufficient conditions for the polyhedral second-order conic form of this inequality to be valid. Then we use the aforementioned n-step conic MIR facet to derive the n-step conic MIR inequality for a general PSOC set and provide conditions for it to be facet-defining. These inequalities are generated using functions which we refer to as the n-step conic MIR functions. We further show that the n-step conic MIR inequality for a general PSOC set strictly dominates the n-step MIR inequalities written for the two linear constraints that define the PSOC set. We also prove that the n-step MIR inequality for a linear mixed integer constraint is a special case of the n-step conic MIR inequality.