Randomized Quasi-Monte Carlo Methods for Risk-Averse Stochastic Optimization

Abstract We establish epigraphical and uniform laws of large numbers for sample-based approximations of law invariant composite risk functionals. These sample-based approximation schemes include Monte Carlo and certain randomized quasi-Monte Carlo integration methods, such as scrambled net integration. Our results can be applied to the approximation of risk-averse stochastic programs and risk-averse stochastic variational inequalities. Our numerical simulations empirically demonstrate that randomized quasi-Monte Carlo approaches based on scrambled Sobol’ sequences can yield smaller bias and root mean square error than Monte Carlo methods for risk-averse optimization.