We study two-stage stochastic mixed integer programs (TSS-MIPs) with integer variables in the second stage. We show that under suitable conditions, the second stage MIPs can be convexified by adding parametric cuts a priori. As special cases, we extend the results of Miller and Wolsey (Math Program 98(1):73-88, 2003) to TSS-MIPs. Furthermore, we consider second stage programs that are generalizations of the well-known mixing (and continuous mixing) set, or certain piecewise-linear convex objective integer programs. These results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the TSS-MIP. We also use four variants of the two-stage stochastic capacitated lot-sizing problems as test problems for computational experiments, and present tight second-stage formulations for these problems. Our computational results show that adding parametric inequalities that a priori convexify the second stage formulation significantly reduces the total solution time taken to solve these problems.
M. Bansal, K. L. Huang, S. Mehrotra, Tight second-stage formulations for two-stage stochastic mixed integer programming problems," SIAM Journal on Optimization, 28(1), 788-819, 2018.