| Abstract: | Discrete approximation, which has been the prevailing scheme in stochastic programming in the past decade, has been extended to distributionally robust optimization (DRO) recently. In this paper, we conduct rigorous quantitative stability
analysis of discrete approximation schemes for DRO, which measures the approximation error in terms of discretization sample size. For the ambiguity set defined
through equality and inequality moment conditions, we quantify the discrepancy between the discretized ambiguity sets and the original set with respect to the Wasserstein metric. To establish the quantitative convergence, we develop a Hoffman error
bound theory with Hoffman constant calculation criteria in a infinite dimensional
space, which can be regarded as a byproduct of independent interest. For the ambiguity set defined by Wasserstein ball and moment conditions combined with Wasserstein ball, we present similar quantitative stability analysis by taking full advantage
of the convex property inherently admitted by Wasserstein metric. Efficient numerical methods for specifically solving discrete approximation DRO problems with
thousands of samples are also designed. In particular, we reformulate different types
of discrete approximation problems into a class of saddle point problems with completely separable structures. The stochastic primal-dual hybrid gradient (PDHG) algorithm where in each iteration we update a random subset of the sampled variables
is then amenable as a solution method for the reformulated saddle point problems.
Some preliminary numerical tests are reported. |
