Multistage Stochastic Optimization via Kernels
This provides a tractable method for solving complex stochastic optimization problems in domains like inventory management, though it appears incremental as it builds on kernel methods with sparsification.
The paper tackles multistage stochastic optimization problems where decisions do not affect uncertainty by developing a non-parametric, data-driven approach using reproducing kernel Hilbert spaces and functional stochastic gradient descent, achieving near-optimal average performance in computational experiments on stochastic inventory management.
We develop a non-parametric, data-driven, tractable approach for solving multistage stochastic optimization problems in which decisions do not affect the uncertainty. The proposed framework represents the decision variables as elements of a reproducing kernel Hilbert space and performs functional stochastic gradient descent to minimize the empirical regularized loss. By incorporating sparsification techniques based on function subspace projections we are able to overcome the computational complexity that standard kernel methods introduce as the data size increases. We prove that the proposed approach is asymptotically optimal for multistage stochastic optimization with side information. Across various computational experiments on stochastic inventory management problems, {our method performs well in multidimensional settings} and remains tractable when the data size is large. Lastly, by computing lower bounds for the optimal loss of the inventory control problem, we show that the proposed method produces decision rules with near-optimal average performance.