Moment Convergence Rate in Stochastic Optimization

where X is the decision set and Q0, the distribution of ξ, is a probability distribution supported on [0, 1]. This kind of problem can be solved numerically or (in special cases) in closed form if we know the exact distribution of Q0. Unfortunately, in practice one rarely knows the exact distribution Q0. Instead, one often has only partial information about Q0, which may include limited distributional information such as a few moments or quantiles. In a seminal paper, Scarf proposed and solved a “robust version” of Problem (1.1). More precisely, suppose that instead of knowing Q0 exactly, one only knew the first two moments of Q0, and wanted to optimize against an adversary who could pick a “worst-case” distribution subject to those moment constraints. Then one is faced with the min-max optimization problem