First order asymptotics of the sample average approximation method to solve risk averse stochastic progams

TL;DR

This paper establishes first-order asymptotic results for the sample average approximation method in stochastic programming, introducing new empirical process conditions that do not require pathwise continuity, applicable to risk-averse and risk-neutral problems.

Contribution

It introduces novel empirical process conditions for analyzing SAA optimal values without assuming pathwise continuity, extending results to risk-averse stochastic programs.

Findings

Derived CLT-type results for SAA optimal values.

New conditions applicable to H"older continuous goal functions.

Extended analysis to risk-averse stochastic programs.

Abstract

We investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study in Kr\"atschmer (2023). Central Limit Theorem type results are derived for the optimal value. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are H\"older continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise H\"older continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the Sample Average Approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and divergence risk measures.