Approximation-theoretic computational methods for scalable decision optimization in complex industrial engineering systems

Abstract

This paper presents an approximation-theoretic computational framework designed to resolve the curse of dimensionality in non-convex, high-dimensional stochastic optimization problems common to modern industrial systems like dynamic supply chains and power grids. The framework constructs dynamic Chebyshev-polynomial surrogate approximations of the Bellman value function over a compact hypercube domain, integrated with Wasserstein-based distributionally robust chance constraints that reduce to tractable conic relaxations through optimal transport strong duality. Theoretical guarantees include a Jackson-type uniform convergence theorem for the truncated tensor-product Chebyshev expansion and a stability lemma bounding quadrature perturbations using the Chebyshev nodes' Lebesgue constant. Incorporating Smolyak sparse grids significantly reduces node evaluation count while preserving Jackson-type error bounds up to a logarithmic factor. An adaptive Chebyshev value-iteration algorithm is developed for finite-horizon decision problems and validated on a 48-state multi-echelon supply chain and a smart-grid energy storage dispatch case. Benchmarked against full-grid dynamic programming, stochastic dual dynamic programming (SDDP), and deep deterministic policy gradient (DDPG) on a 24-dimensional instance, the proposed approach reduces computation time from 890.2–1,240.0 seconds to 42.6 seconds and cuts memory usage from 420–850 MB to 110 MB. It achieves a relative approximation error of 10−4 to 10−3, substantially outperforming baseline error rates of 10−2 to 10−1. At a 48-dimensional scale, the method maintains scalability with a 184.3-second compute time and an error of 10−3 to 10−2, remaining strictly under the target tolerance. These results confirm polynomial-time scaling and order-of-magnitude acceleration over established industrial solvers while maintaining rigorous error control.