Randomized quasi-Monte Carlo methods for risk-averse stochastic optimization

TL;DR

This paper develops theoretical laws for approximating risk functionals using Monte Carlo and randomized quasi-Monte Carlo methods, demonstrating that RQMC can outperform MC in risk-averse stochastic optimization.

Contribution

It establishes new convergence laws for RQMC methods in risk-averse stochastic optimization and shows empirical advantages over traditional Monte Carlo approaches.

Findings

RQMC methods have smaller bias than MC.

RQMC yields lower root mean square error.

Numerical results confirm RQMC's improved performance.

Abstract

We establish epigraphical and uniform laws of large numbers for sample-based approximations of law invariant risk functionals. These sample-based approximation schemes include Monte Carlo (MC) and certain randomized quasi-Monte Carlo integration (RQMC) 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 RQMC approaches based on scrambled Sobol' sequences can yield smaller bias and root mean square error than MC methods for risk-averse optimization.