In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of \(l_p\)-norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. \(p=1\) and \(p =2\), respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts.

在本文中，我们使用Atamtürk和Narayanan的圆锥混合整数舍入（CMIR）割生成程序，推导了具有多个整数变量的确定性多约束多面圆锥混合整数集的（部分）凸包（Math Prog 122：1–20， 2008年），从而将其结果扩展为具有单约束和一个整数变量的简单多面圆锥混合整数集。然后，我们介绍两阶段随机p阶圆锥混合整数程序（用TSS-CMIPs表示），其中第二阶段问题的总和为\（l_p \）目标函数中的-范数以及整数变量。首先，我们提出了充分的条件，在这些条件下，在扩展的TSS-CMIPs公式中添加基于场景的非线性削减足以放松对第二阶段整数变量的完整性限制，而不会影响TSS-CMIP最优解的完整性。在第二阶段，我们将基于方案的CMIR割用于TSS-CMIP及其结构化CMIP的分布稳健概括，并证明这些割为第二阶段CMIP提供了圆锥/线性编程等效项或近似值。\（p = 1 \）和\（p = 2 \）。我们观察到，添加基于场景的削减后，解决这些问题所花费的总时间大大减少了。