Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

Abstract

We analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program's solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs. We present numerical results illustrating our theoretical findings.

20 pages, 4 figures

References in corpus (4)

• A framework for automated PDE-constrained optimisation
• Stochastic Proximal Gradient Methods for Nonconvex Problems in Hilbert Spaces
• A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty
• Asymptotic Properties of Monte Carlo Methods in Elliptic PDE-Constrained Optimization under Uncertainty

Cited by in corpus (5)

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• Sample Size Estimates for Risk-Neutral Semilinear PDE-Constrained Optimization
• Reliable Error Estimates for Optimal Control of Linear Elliptic PDEs with Random Inputs
• Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces
• Consistency of sample-based stationary points for infinite-dimensional stochastic optimization