Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty

TL;DR

This paper introduces a Rockafellian relaxation framework for PDE-constrained stochastic optimization that enhances robustness to distributional uncertainty, enabling better outlier detection, removal, and variance reduction.

Contribution

It develops a novel relaxation approach using a bivariate objective functional with a perturbation variable to handle distributional ambiguity in PDE-constrained optimization.

Findings

Framework converges to uncorrupted objectives as corruption vanishes

Numerical examples demonstrate effectiveness in outlier detection and variance reduction

Method improves robustness against distributional inaccuracies in stochastic PDE problems

Abstract

Stochastic optimization problems are generally known to be ill-conditioned to the form of the underlying uncertainty. A framework is introduced for optimal control problems with partial differential equations as constraints that is robust to inaccuracies in the precise form of the problem uncertainty. The framework is based on problem relaxation and involves optimizing a bivariate, "Rockafellian" objective functional that features both a standard control variable and an additional perturbation variable that handles the distributional ambiguity. In the presence of distributional corruption, the Rockafellian objective functionals are shown in the appropriate settings to $\Gamma$-converge to uncorrupted objective functionals in the limit of vanishing corruption. Numerical examples illustrate the framework's utility for outlier detection and removal and for variance reduction.