A simulation-based approach to two-stage stochastic programming with recourse
Mathematical Programming: Series A and B
Variable-sample methods for stochastic optimization
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication
SIAM Journal on Computing
Adaptive Newton-based multivariate smoothed functional algorithms for simulation optimization
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Recourse-based stochastic nonlinear programming: properties and Benders-SQP algorithms
Computational Optimization and Applications
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This paper presents a finite dimensional approach to stochastic approximation in infinite dimensional Hilbert space. The problem was motivated by applications in the field of stochastic programming wherein we minimize a convex function defined on a Hilbert space. We define a finite dimensional approximation to the Hilbert space minimizer. A justification is provided for this finite dimensional approximation. Estimates of the dimensionality needed are also provided. The algorithm presented is a two time-scale Newton-based stochastic approximation scheme that lives in this finite dimensional space. Since the finite dimensional problem can be prohibitively large dimensional, we operate our Newton scheme in a projected, randomly chosen smaller dimensional subspace.