Gaussian Process (GP) models are widely utilized as surrogate models in scientific and engineering fields. However, standard GP models are limited to continuous variables due to the difficulties in establishing correlation structures for categorical variables. To overcome this limitation, we introduce \textbf{WE}ighted Euclidean distance matrices \textbf{G}aussian \textbf{P}rocess (WEGP). WEGP constructs the kernel function for each categorical input by estimating the Euclidean distance matrix (EDM) among all categorical choices of this input. The EDM is represented as a linear combination of several predefined base EDMs, each scaled by a positive weight. The weights, along with other kernel hyperparameters, are inferred using a fully Bayesian framework. We analyze the predictive performance of WEGP theoretically. Numerical experiments validate the accuracy of our GP model, and by WEGP, into Bayesian Optimization (BO), we achieve superior performance on both synthetic and real-world optimization problems. The code is available at: \url{https://github.com/pmy0124nus/WEGP.}

In practice, objective functions of real-time control systems can have multiple local minimums or can dramatically change over the function space, making them hard to optimize. To efficiently optimize such systems, in this paper, we develop a parallel global optimization framework that combines direct search methods with parallel Bayesian optimization. It consists of an iterative global and local search that searches broadly through the entire global space for promising regions and then efficiently exploits each local promising region. We prove the asymptotic convergence properties of the proposed framework and conduct several numerical experiments to illustrate its empirical performance. We also provide a real-time control problem to illustrate the efficiency of our proposed algorithm.Note to Practitioners—This work is motivated by a collision avoidance problem of vessels aided with onboard agent-based simulations. The simulation on one vessel can predict potential conflicts with other vessels on a pre-defined trajectory. In heavy congestion regions, the environment is highly dynamic and thus it is difficult to find a much safer alternative trajectory if collision is predicted on the current one. Moreover, for such real-time decisions, the control system should be quick in response to improve safety. The proposed metamodel based algorithm is designed for quick decision in such highly dynamic systems. The algorithm employs a decomposition of the response surface to better handle the multi-modal surface resulting from the highly dynamic environment. Specifically, it first looks at the large-scale trend globally (filter out the many local fluctuations that may otherwise trap the algorithm) to locate potential promising regions and then proceeds to this local regions for more detailed local search. To make quick decisions, it uses fast direct search algorithms in the local search phase and applies a parallel search scheme to enjoy the abundant computing power. Both the theoretical analysis and the simulation studies demonstrate that the proposed algorithm can provide better decisions quickly. We also note that this algorithm is not limited to real-time control or simulation-based system. In the case where each run of the experiment is expensive and the budget is limited for the final decision and when the response function is multi-modal, this algorithm can hopefully become a quite efficient and competitive approach. The multi-modal responses have broad applications in the area of control, planning and operations research, such as robot navigating and reinforcement learning.

Gaussian process (GP) model based optimization is widely applied in simulation and machine learning. In general, it first estimates a GP model based on a few observations from the true response and then uses this model to guide the search, aiming to quickly locate the global optimum. Despite its successful applications, it has several limitations that may hinder its broader use. First, building an accurate GP model can be difficult and computationally expensive, especially when the response function is multimodal or varies significantly over the design space. Second, even with an appropriate model, the search process can be trapped in suboptimal regions before moving to the global optimum because of the excessive effort spent around the current best solution. In this work, we adopt the additive global and local GP (AGLGP) model in the optimization framework. The model is rooted in the inducing points based GP sparse approximations and is combined with independent local models in different regions. With these properties, the AGLGP model is suitable for multimodal responses with relatively large data sizes. Based on this AGLGP model, we propose a combined global and local search for optimization (CGLO) algorithm. It first divides the whole design space into disjoint local regions and identifies a promising region with the global model. Next, a local model in the selected region is fit to guide detailed search within this region. The algorithm then switches back to the global step when a good local solution is found. The global and local natures of CGLO enable it to enjoy the benefits of both global and local search to efficiently locate the global optimum. Summary of Contribution: This work proposes a new Gaussian process based algorithm for stochastic simulation optimization, which is an important area in operations research. This type of algorithm is also regarded as one of the state-of-the-art optimization algorithms for black-box functions in computer science. The aim of this work is to provide a computationally efficient optimization algorithm when the baseline functions are highly nonstationary (the function values change dramatically across the design space). Such nonstationary surfaces are very common in reality, such as the case in the maritime traffic safety problem considered here. In this problem, agent-based simulation is used to simulate the probability of collision of one vessel with the others on a given trajectory, and the decision maker needs to choose the trajectory with the minimum probability of collision quickly. Typically, in a high-congestion region, a small turn of the vessel can result in a very different conflict environment, and thus the response is highly nonstationary. Through our study, we find that the proposed algorithm can provide safer choices within a limited time compared with other methods. We believe the proposed algorithm is very computationally efficient and has large potential in such operational problems.

A quantile is a popular performance measure for a stochastic system to evaluate its variability and risk. To reduce the risk, selecting the actions that minimize the tail quantiles of some loss distributions is typically of interest for decision makers. When the loss distribution is observed via simulations, evaluating and optimizing its quantile can be challenging, especially when the simulations are expensive as it may cost a large number of simulation runs to obtain accurate quantile estimators. In this work, we propose a multilevel metamodel (cokriging)-based algorithm to optimize quantiles more efficiently. Utilizing nondecreasing properties of quantiles, we first search on cheaper and informative lower quantiles, which are more accurate and easier to optimize. The quantile level iteratively increases to the objective level, and the search has a focus on the possible promising regions identified by the previous levels. This enables us to leverage the accurate information from the lower quantiles to find the optimums faster and improve algorithm efficiency.