Supermodularity and robust optimization in elliptic PDEs

Abstract

We study distributionally robust optimization problems constrained by semilinear elliptic PDEs. Uncertain nonnegative loads enter the PDE through $m$ intensity parameters, and damage is measured by integrating a nondecreasing convex function of the solution. We prove that, for coercive operators satisfying a positivity-preserving lattice condition, a nondecreasing concave reaction $g$ makes the damage nondecreasing, convex, and supermodular in the load intensities. Under nonnegative loading and $g(0)=0$, concavity is also necessary for these properties to hold for all admissible operators and damage functions. For Wasserstein uncertainty sets based on a weighted $\ell_1$ distance and $N$ observations, we show that at most $m+1$ nested stress corners are needed per observation. These corners form a set of at most $N(m+1)$ points that does not depend on the Wasserstein radius and supports a worst-case distribution for every radius. We compute the exact robust value by checking $λ=0$ and at most $Nm$ positive breakpoints, and we construct a worst-case distribution that moves each observation to a chain corner and splits at most one observation between two adjacent corners. We apply the results to a convex remediation-planning problem with Monod biodegradation. Numerical experiments show that the method requires far fewer PDE solves than enumerating all corners. Finally, a counterexample shows that supermodularity can fail for the fractional $p$-Laplacian when $p>2$.