Epi-convergent discretizations of stochastic programs via integration quadratures

The simplest and the best-known method for numerical approximation of high-dimensional integrals is the Monte Carlo method (MC), i.e. random sampling. MC has also become the most popular method for constructing numerically solvable approximations of stochastic programs. However, certain modern integration quadratures are often superior to crude MC in high-dimensional integration, so it seems natural to try to use them also in discretization of stochastic programs. This paper derives conditions that guarantee the epi-convergence of the resulting objectives to the original one. Our epi-convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure. As examples, we prove epi-convergence of quadrature-based discretizations of three different models of portfolio management and we study their behavior numerically. Besides MC, our discretizations are the only existing ones with guaranteed epi-convergence for these problem classes. In our tests, modern quadratures seem to result in faster convergence of optimal values than MC.