Randomized QMC Methods for Mixed-Integer Two-Stage Stochastic Programs with Application to Electricity Optimization

We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second stage. It is known that the second-stage optimal value function is piecewise linear-quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets. This structure is exploited to provide conditions implying that first and second order ANOVA terms of the integrand have mixed first order partial derivatives in the sense of Sobolev. This shows that the integrand can be decomposed into a smooth part and a not well-behaved but small part if the effective dimension is low. This leads to good convergence properties of randomized QMC methods. In a case study we consider an optimization model for generating and trading electricity under normal load and price stochasticity. Our numerical experiments where we compare Monte Carlo and two randomized QMC methods indicate that the latter can be superior which confirms our analysis.