Gradient-based Simulation Optimization using Approximated Systems

Abstract

We propose gradient-based simulation optimization algorithms for complicated stochastic systems, for which the exact simulation of the objective function is extremely costly or impossible. For such complicated stochastic systems, a sequence of systems can often be constructed to approximate the original complicated stochastic system, which has finer and finer approximation resolution but higher and higher cost to simulate. With the goal of optimizing the original system, our proposed algorithms strategically and sequentially utilize the approximated systems to construct stochastic gradients and perform gradient search in the decision space. Various gradient construction methods are analyzed and can accommodate different features of the approximated systems such as discontinuity. As a theory support, we prove algorithm convergence, the convergence rate, central limit theorem, and optimal algorithm design under the assumption that the objective function for original system is strong convex, while no such assumptions on the approximated systems are required. Moreover, if the sequence of approximated systems can be constructed in a coupled way, we develop multi-level gradient-based optimization algorithms. We prove theoretically and then show empirically that the additional use of multi-level structure can further improve the convergence rate and computational efficiency of the proposed simulation optimization algorithms.