We introduce a solution methodology for risk-neutral and risk-averse stochastic programs with deterministic constraints. Our approach relies on principles from projected gradient descent and sample average approximation algorithms. However, we adaptively control the sample size used in computing the reduced gradient approximation at each iteration. This leads to a significant reduction in cost. Numerical experiments from finance and engineering illustrate the performance and efficacy of the presented algorithms.
We introduce adaptive sampling methods for stochastic programs with deterministic constraints. First, we propose and analyze a variant of the stochastic projected gradient method, where the sample size used to approximate the reduced gradient is determined on-the-fly and updated adaptively. This method is applicable to a broad class of expectation-based risk measures, and leads to a significant reduction in the individual gradient evaluations used to estimate the objective function gradient. Numerical experiments with expected risk minimization and conditional value-at-risk minimization support this conclusion, and demonstrate practical performance and efficacy for both risk-neutral and risk-averse problems. Second, we propose an SQP-type method based on similar adaptive sampling principles. The benefits of this method are demonstrated in a simplified engineering design application, featuring risk-averse shape optimization of a steel shell structure subject to uncertain loading conditions and model uncertainty.
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