Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs

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