Improving Sample Average Approximation Using Distributional Robustness

We consider stochastic optimization problems in which we aim to minimize the expected value of an objective function with respect to an unknown distribution of random parameters. We analyse the out-of-sample performance of solutions obtained by solving a distributionally robust version of the sample average approximation problem for unconstrained quadratic problems and derive conditions under which these solutions are improved in comparison with those of the sample average approximation. We compare different mechanisms for constructing a robust solution: phi-divergence using both total variation and standard smooth $\varphi$ functions; a CVaR-based risk measure; and a Wasserstein metric.