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Computational Optimization and Applications (2023) 85:441–478

https://doi.org/10.1007/s10589-023-00461-8

A relaxation-based probabilistic approach for

PDE-constrained optimization under uncertainty with

pointwise state constraints

Drew P. Kouri1·Mathias Staudigl2·Thomas M. Surowiec3

Received: 23 May 2022 / Accepted: 3 February 2023 / Published online: 27 February 2023

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023

Abstract

We consider a class of convex risk-neutral PDE-constrained optimization problems

subject to pointwise control and state constraints. Due to the many challenges asso-

ciated with almost sure constraints on pointwise evaluations of the state, we suggest

a relaxation via a smooth functional bound with similar properties to well-known

probability constraints. First, we introduce and analyze the relaxed problem, discuss

its asymptotic properties, and derive formulae for the gradient the adjoint calculus.

We then build on the theoretical results by extending a recently published online con-

vex optimization algorithm (OSA) to the inﬁnite-dimensional setting. Similar to the

regret-based analysis of time-varying stochastic optimization problems, we enhance

the method further by allowing for periodic restarts at pre-deﬁned epochs. Not only

does this allow for larger step sizes, it also proves to be an essential factor in obtain-

ing high-quality solutions in practice. The behavior of the algorithm is demonstrated

in a numerical example involving a linear advection–diffusion equation with random

inputs. In order to judge the quality of the solution, the results are compared to those

arising from a sample average approximation (SAA). This is done ﬁrst by comparing

the resulting cumulative distributions of the objectives at the optimal solution as a

function of step numbers and epoch lengths. In addition, we conduct statistical tests

to further analyze the behavior of the online algorithm and the quality of its solutions.

BThomas M. Surowiec

thomasms@simula.no

Drew P. Kouri

dpkouri@sandia.gov

Mathias Staudigl

m.staudigl@maastrichtuniversity.nl

1Sandia National Laboratories, P.O. Box 5800, MS-1320, Albuquerque, NM, USA

2Department of Advanced Computing Sciences (DACS), Maastricht University, Maastricht, The

Netherlands

3Department of Numerical Analysis and Scientiﬁc Computing, Simula Research Laboratory, Kristian

Augusts Gate 23, 0164 Oslo, Norway

123

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