Stability and Sample-Based Approximations of Composite Stochastic Optimization Problems

Optimization under uncertainty and risk is ubiquitous in business, engineering, and finance. Typically, we use observed or simulated data in our decision models, which aim to control risk, and result in composite risk functionals. The paper addresses the stability of the decision problems when the composite risk functionals are subjected to measure perturbations at multiple levels of potentially different nature. We analyze data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions and establish laws of large numbers and consistency of the optimal values and solutions. This is the first study to propose and analyze smoothing in data-driven composite optimization problems. It is shown that kernel-based and wavelet estimation provide less biased estimation of the risk compared with the empirical plug-in estimators under some assumptions.