Rockafellian Relaxation for PDE-Constrained Optimization with Distributional Uncertainty

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 $Γ$-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.

💡 Research Summary

The paper addresses the instability of stochastic PDE‑constrained optimization when the underlying probability distribution is misspecified or corrupted. Traditional distributionally robust optimization (DRO) mitigates this by minimizing the worst‑case expected value over a set of plausible measures, but this approach is often overly conservative and computationally demanding. The authors propose an optimistic alternative called distributionally optimistic optimization (DOO) based on a Rockafellian relaxation.

A Rockafellian is a bivariate functional Φ(z, t) that coincides with the original objective φ(z) when the perturbation variable t equals zero. The authors construct Φ by adding a perturbation of the probability measure (through a vector t) and a quadratic regularization term θ‖t‖², together with an indicator enforcing that the perturbed measure remains a probability vector. This formulation allows the optimizer to adjust the underlying distribution within a small neighborhood, effectively identifying and neutralizing outlying mass that would otherwise distort the solution.

The core theoretical contribution is a Γ‑convergence analysis in the strong topology of Banach spaces (including Sobolev spaces typical for PDEs). Under mild measurability, weak‑strong continuity of the solution operator s(ξ, z), and lower‑semicontinuity of the cost functions f₀ and g, the authors prove that the family of Rockafellian functionals Φ_ε Γ‑converges to the original functional φ as the corruption magnitude ε→0. Two corruption scenarios are treated: (i) perturbations of a probability density function and (ii) perturbations of the support of a probability measure. In the finite‑dimensional case (sample‑average approximation), they also establish Mosco convergence, which implies Γ‑convergence and guarantees convergence of minimizers.

A simple one‑dimensional example illustrates the phenomenon: a tiny probability ε placed on a large outlier value flips the minimizer from x = 1 to x = 0, yet the Rockafellian relaxation recovers the original minimizer by selecting an appropriate t.

The theory is then applied to stochastic elliptic PDEs with random boundary conditions or coefficients. The solution operator satisfies the required continuity properties via the Lax‑Milgram theorem, and the functional setting is H¹₀(Ω). Numerical experiments demonstrate two practical benefits: (a) outlier detection and removal, leading to a substantial reduction in the variance of the optimal control, and (b) robustness to corrupted probability densities, where the relaxed problem yields controls close to those obtained with the true distribution. The regularization parameter θ plays a crucial role: too small a value leads to numerical instability, while too large a value prevents effective correction of the distribution.

From a computational standpoint, the Rockafellian adds only a low‑dimensional perturbation variable t (typically the number of probability categories) to the original control variable. The resulting problem can be tackled with alternating minimization or ADMM, reusing existing PDE‑constrained optimization solvers for each subproblem, thus keeping the overhead modest.

In conclusion, the paper introduces a novel, mathematically rigorous framework for handling distributional ambiguity in PDE‑constrained optimization. By embedding the original problem in a family of perturbed problems via a Rockafellian, it achieves Γ‑convergence to the uncorrupted objective, ensures convergence of minimizers, and provides practical tools for outlier mitigation and variance reduction. The work opens several avenues for future research, including extensions to nonlinear PDEs, integration with data‑driven probability model learning, and multi‑objective formulations.