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 ℓ1 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.
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BibTeX
@misc{mukherjee2026supermodularity,
title = {{Supermodularity and robust optimization in elliptic PDEs}},
author = {Debangana Mukherjee and Akash Ashirbad Panda},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02573},
eprint = {2611.02573},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02573},
}
RIS
TY - UNPB
TI - Supermodularity and robust optimization in elliptic PDEs
AU - Debangana Mukherjee
AU - Akash Ashirbad Panda
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02573
UR - https://arxiv.org/abs/2611.02573
AB - 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 ℓ_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.
ER -