Quasi-Monte Carlo methods for two-stage stochastic mixed-integer programs

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 higher order terms of the integrand's ANOVA decomposition (Math. Comp. 79 (2010), 953-966) have mixed weak first order partial derivatives. This leads to a good smooth approximation of the integrand and, hence, to good convergence rates of randomized QMC methods if the effective (superposition) dimension is low.

Automated summary

This study investigates the use of randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems involving mixed-integer decisions. By exploiting the piecewise linear-quadratic structure of the second-stage optimal value function, the study establishes conditions under which randomized QMC methods exhibit good convergence rates, particularly when the effective dimension is low.