Stochastic Localization Methods for Convex Discrete Optimization via Simulation

Abstract

We develop and analyze a set of new sequential simulation-optimization algorithms for large-scale multi-dimensional discrete optimization via simulation problems with a convexity structure. The "large-scale" notion refers to that the decision variable has a large number of values to choose from on each dimension. The proposed algorithms are targeted to identify a solution that is close to the optimal solution given any precision level with any given probability. To achieve this target, utilizing the convexity structure, our algorithm design does not need to scan all the choices of the decision variable, but instead sequentially draws a subset of choices of the decision variable and uses them to "localize" potentially near-optimal solutions to an adaptively shrinking region. To show the power of the localization operation, we first consider one-dimensional large-scale problems. We propose the shrinking uniform sampling algorithm, which is proved to achieve the target with an optimal expected simulation cost under an asymptotic criterion. For multi-dimensional problems, we combine the idea of localization with subgradient information and propose a framework to design stochastic cutting-plane methods and the dimension reduction algorithm, whose expected simulation cost have a low dependence on the scale and the dimension of the problems. The proposed algorithms do not require prior information about the Lipschitz constant of the objective function and the simulation costs are upper bounded by a value that is independent of the Lipschitz constant. Finally, we propose an adaptive algorithm to deal with the unknown noise variance case under the assumption that the randomness of the system is Gaussian. We implement the proposed algorithms on both synthetic and queueing simulation optimization problems, and demonstrate better performances compared to benchmark methods.