Efficient discrete optimization via simulation using stochastic kriging

Abstract

We propose to use a global metamodeling technique known as stochastic kriging to improve the efficiency of Discrete Optimization-via-Simulation (DOvS) algorithms. Stochastic kriging metamodel allows the DOvS algorithm to utilize all information collected during the optimization process and identify solutions that are most likely to lead to significant improvement in solution quality. We call the approach Stochastic Kriging for OPtimization Efficiency (SKOPE). In this paper, we integrate SKOPE with a locally convergent DOvS algorithm known as Adaptive Hyperbox Algorithm (AHA). Numerical experiments show that SKOPE significantly improves the performance of AHA in the early stage of optimization, which is very helpful for DOvS applications where the number of simulations for an optimization task is severely limited due to a short decision time window and time-consuming simulation. INTRODUCTIONIn the past decade, there has been a fast growing body of literature on how to optimize the design of a system using a simulation model. We refer to such problems as Optimization via Simulation (OvS). See Fu (2002) andFu, Glover, andApril (2005) for a review of OvS. When the decision variables are discrete valued, such as the stocking levels of products in a multi-product inventory management problem, we refer to it as Discrete Optimization via Simulation (DOvS).When there are only a few hundreds feasible solutions, ranking-and-selection algorithms can be applied to choose the best solution. Examples include the indifference-zone procedure of Nelson et al. (2001) and Kim and Nelson (2001), the Bayesian procedure of Chick and Inoue (2001), and the Optimal Computing Budget Allocation (OCBA) procedure of Chen et al. (2000). Most recently, Frazier (2012) proposed an indifference-zone procedure for more than 15,000 alternatives. However, in a typical DOvS problem, the feasible solution space often includes millions and even billions of feasible solutions and thus rankingand-selection procedures are not directly applicable.Adaptive random search has been the dominant paradigm for designing DOvS algorithms when the solution space is large. Most existing DOvS algorithms focus on asymptotic global convergence, including the stochastic ruler algorithm of Yan and Mukai (1992), the simulated annealing algorithm of Alrefaei and Andradóttir (1999), and the nested partitions algorithm of Shi and Ólafsson (2000) and Pichitlamken and Nelson (2003). These algorithms essentially have to visit every solution to guarantee global convergence and lack the efficiency necessary to solve real world problems.Another class of adaptive random search DOvS algorithms guarantees convergence to a local optimal solution. Andradóttir (1995) proposed a locally convergent algorithm for one-dimensional DOvS problems. The COMPASS algorithm of Hong and Nelson (2006) and the AHA algorithm of Xu, Hong, and Nelson (2011) also belong to this class of DOvS algorithms, but they are not restricted to one dimension. By focusing on finding a local optimum, these algorithms ca...

Comparison of Kriging-Based Methods for Simulation Optimization with Heterogeneous Noise

“…For the experiments in Section 4 , we implemented this algorithm on a server with 5 cores and 100 threads, running each macroreplication on a different thread. However, in many problems, the feasible region contains millions of points ( Xu, 2012 ); the computational requirements may limit the applicability of CKG in these cases. …”

“…Many methods are available for solving problem (1) with a finite set of points such as the ranking and selection (R&S) method (see Hong, Nelson, and Xu, 2015 , for a review). Most of them are, however, unsuitable when (1) the number of feasible solutions is large, or (2) simulation replications are timeconsuming ( Xu, 2012 ). Kriging-based or Bayesian optimization is among the few techniques that can handle this problem with low problem dimensionality ( d ≤ 20, Brochu, Cora, & De Freitas, 2010;Preuss, Wagner, & Ginsbourger, 2012 ).…”

A Review of Random Search Methods

“…Hong and Nelson [46] prove convergence w.p.1 to a local optimal solution and provide numerical examples. This work is continued by Hong [45], Hong et al [47], and Xu et al [78,79] who improve the efficiency of the original COMPASS approach (see also the discussion of Xu [77] in Sect. 10.4.1 below).…”

“…Huang et al [51], Sun et al [74], and Xu [77] all use Kriging meta-models and random search to solve simulation optimization problems. More specifically, Huang et al [51] propose the SKO (Sequential Kriging Optimization) approach, where each iteration starts with a kriging meta-model of the objective function, identifies a solution that maximizes an Expected Improvement (EI) function (described in Sect.…”

Simulation Optimization: A Review and Exploration in the New Era of Cloud Computing and Big Data

“…Stochastic kriging (Ankenman et al, 2010) extends kriging to stochastic simulation with unequal variances. In a recent study, a stochastic kriging metamodel is integrated with a locally convergent black-box search method to improve its efficiency (Xu, 2012). In general, using a stochastic kriging metamodel can support both continuous and discrete-valued decision variables.…”