Risk-averse optimization under distributional uncertainty with Rockafellian relaxation

TL;DR

This paper introduces a risk-averse optimization framework resilient to distributional ambiguities, applicable to PDE-constrained problems, combining DRO and DOO approaches with novel theoretical results.

Contribution

It advances the theory with strengthened convergence, existence, and optimality results, accommodating infinite-dimensional probability spaces without finite-noise assumptions.

Findings

Framework enhances out-of-sample performance.

Addresses adversarial and outlier data effectively.

Applicable to PDE-constrained problems with distributional uncertainty.

Abstract

A framework for risk-averse optimization problems is introduced that is resilient to ambiguities in the true form of the underlying probability distribution. The focus is on problems with partial differential equations (PDEs) as constraints, although the formulation is more broadly applicable. The framework is based on combining risk measures with problem relaxation techniques, and it builds off of previous advances for risk-neutral problems. This work advances the existing theory with strengthened $\Gamma$-convergence results, novel existence results and first-order optimality criteria. In particular, the theoretical approach naturally accommodates infinite-dimensional probability spaces; no finite-dimensional noise assumption is needed. The framework blends aspects of both distributionally robust optimization (DRO) and distributionally optimistic optimization (DOO) approaches. The DRO aspect facilitates strong out-of-sample performance, while the DOO aspect takes care of adversarial and outlier data, as illustrated with numerical examples.