Bayesian Simulation Optimization with Common Random Numbers

This paper addresses discrete optimization via simulation. We show that allowing for both a correlated prior distribution on the means (e.g., with discrete Kriging models) and sampling correlation (e.g., with common random numbers, or CRN) can significantly improve the ability to quickly identify the best alternative. These two correlations are brought together for the first time in a highly sequential knowledge-gradient sampling algorithm, which chooses points to sample using a Bayesian value of information (VOI) criterion. We provide almost sure convergence guarantees as the number of samples grows without bound when parameters are known and provide approximations that allow practical implementation including a novel use of the VOI’s gradient rather than the response surface’s gradient. We demonstrate that CRN leads to improved optimization performance for VOI-based algorithms in sequential sampling environments with a combinatorial number of alternatives and costly samples.

We present Q-SASS, a quasi-Newton method for unconstrained stochastic optimization that does not rely on common random numbers. Most existing quasi-Newton approaches leverage common random numbers to construct second-order updates. However, motivated by challenges in variational quantum algorithms—where such coordination is not possible—we consider the setting in which function values and gradients are accessible only through noisy probabilistic zeroth- and first-order oracles, and no common random numbers can be exploited. We derive high-probability tail bounds on the iteration complexity of our algorithm for nonconvex, convex, and strongly convex (more generally, those satisfying the PL condition) objective functions. Finally, we demonstrate the empirical benefits of our quasi-Newton updating scheme on both synthetic and quantum chemistry problems.

We consider noisy optimization and some traditional variance reduction techniques aimed at improving the convergence rate, namely (i) common random numbers (CRN), which is relevant for population-based noisy optimization and (ii) stratified sampling, which is relevant for most noisy optimization problems. We present artificial models of noise for which common random numbers are very efficient, and artificial models of noise for which common random numbers are detrimental. We then experiment on a desperately expensive unit commitment problem. As expected, stratified sampling is never detrimental. Nonetheless, in practice, common random numbers nonetheless provided, by far, most of the improvement.

Contributors

Physical phenomena are often studied using numerical simulators. Such computer codes are a function of uncertain input parameters and a global sensitivity analysis (GSA) can be performed to identify their impacts on the simulator outputs. Sobol' indices, based on output variance decomposition, are commonly used to perform quantitative GSA. For many years now, other tools have been studied, closer to physical practices such as the derivative-based global sensitivity measures (DGSM). However, numerical simulators rarely provide the output gradient and DGSM estimation is not directly possible. To address this limitation, we propose to estimate the DGSMs using a Gaussian process metamodel (GPM) which approximates the simulator. Based on this GPM, we propose two DGSM estimators: a plug-in one defined by the DGSM of the GPM predictor and another one defined by the expectation of the DGSM associated to the full-GPM. The latter is equal to the first one completed by a variance term and can be accompanied by a cre...

This paper deals with the optimization of industrial asset management strategies, whose profitability is characterized by the Net Present Value (NPV) indicator which is assessed by a Monte Carlo simulator. The developed method consists in building a metamodel of this stochastic simulator, allowing to obtain, for a given model input, the NPV probability distribution without running the simulator. The present work is concentrated on the emulation of the quantile function of the stochastic simulator by interpolating well chosen basis functions and metamodeling their coefficients (using the Gaussian process metamodel). This quantile function metamodel is then used to treat a problem of strategy maintenance optimization (four systems installed on different plants), in order to optimize an NPV quantile. Using the Gaussian process framework, an adaptive design method (called quantile function expected improvement) is defined by extending in our case the well‐known efficient global optimization algorithm. This allows to obtain an “optimal” solution using a small number of simulator runs. Copyright © 2016 John Wiley & Sons, Ltd.

Many algorithms have been devised for minimizing the costs associated with obtaining the answer to a single, isolated query in a distributed database system. However, if more than one query may be processed by the system at the same time and if the arrival times of the queries are unknown, the determination of optimal query-processing strategies becomes a stochastic optimization problem. In order to cope with such problems, a theoretical state-transition model is presented that treats the system as one operating under a stochastic load. Query-processing strategies may then be distributed over the processors of a network as probability distributions, in a manner which accommodates many queries over time. It is then shown that the model leads to the determination of optimal query-processing strategies as the solution of mathematical programming problems, and analytical results for several examples are presented. Furthermore, a divide-and-conquer approach is introduced for decomposing stochastic query optimization problems into distinct subproblems for processing queries sequentially and in parallel.

We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of risk averse or risk-prone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). We show that we can obtain a polynomial time approximation algorithm with additive error ϵ for any ϵ >; 0, if there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above) and the maximum value of the utility function is bounded by a constant. Our result generalizes several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.

This paper compares two metamodel-based methodologies for multi-criteria simulation optimization (SO), using the injection molding of a disposable camera front housing as a case study. The first methodology uses linear regression metamodels and data envelopment analysis (DEA) to iteratively identify potential Pareto solutions, while the second one uses a Gaussian process metamodel and calculates an expected improvement to determine the new input runs sequentially. A one-to-one comparison of the approaches is presented using two optimization examples. The first example involves two process outputs and the second involves three. The approaches are evaluated using the same number of simulations, and are compared in terms of the quality of the obtained Pareto front, based on the hypervolume indicator. Advantages and disadvantages of both methods are discussed.

We consider optimization via simulation over a finite set of alternatives. We employ a Bayesian value-ofinformation approach in which we allow both correlated prior beliefs on the sampling means and correlated sampling. Correlation in the prior belief allow us to learn about an alternative’s value from samples of similar alternatives. Correlation in sampling, achieved through common random numbers, allows us to reduce the variance in comparing one alternative to another. We allow for a more general combination of both types of correlation than has been offered previously in the Bayesian ranking and selection literature. We do so by giving an exact expression for the value of information for sampling the difference between a pair of alternatives, and derive new knowledge-gradient methods based on this valuation. 1

We present a new random search method for solving simulation optimization problems. Our approach emphasizes the need for maintaining the right balance between exploration and exploitation during various stages of the search. Exploration represents global search for promising solutions within the entire feasible region, while exploitation involves local search of promising subregions. Preliminary numerical results are provided that show the performance of the method applied to solve deterministic and stochastic optimization problems.

This paper considers data-driven chance-constrained stochastic optimization problems in a Bayesian framework. Bayesian posteriors afford a principled mechanism to incorporate data and prior knowledge into stochastic optimization problems. However, the computation of Bayesian posteriors is typically an intractable problem, and has spawned a large literature on approximate Bayesian computation. Here, in the context of chance-constrained optimization, we focus on the question of statistical consistency (in an appropriate sense) of the optimal value, computed using an approximate posterior distribution. To this end, we rigorously prove a frequentist consistency result demonstrating the convergence of the optimal value to the optimal value of a fixed, parameterized constrained optimization problem. We augment this by also establishing a probabilistic rate of convergence of the optimal value. We also prove the convex feasibility of the approximate Bayesian stochastic optimization problem. Finally, we demonstrate the utility of our approach on an optimal staffing problem for an M/M/c queueing model.

Traditionally, exactness, numerical stability and speed are the three main criteria for evaluating algorithms for random variate generation. However, it is sometimes required that the algorithms provide correlation between generated variates for the purpose of inducing dependence among the output of simulation runs. The inverse transformation, which produces optimal correlation induction, often performs poorly in terms of the first three criteria. Algorithms based on composition, rejection, and special properties which often excel in terms of the first three criteria, tend to scramble the use of random numbers, causing many attempts at common random numbers, antithetic variates and external control variates to fail. The concept of obtaining correlation via algorithms other than the inverse transformation is examined here. To demonstrate feasibility, previously developed algorithms for Poisson and binomial random variate generation are modified to obtain both positive and negative correlation between runs....

Efficient planning of construction operations is deemed necessary to meet project objectives. Researchers have used simulation optimization to select the optimum amount of equipment and number of crews for construction operations. However, the current state of the practice suffers from the long computation time and the presence of inferior solutions in the final Pareto front. The objective of this paper is to develop and evaluate a robust simulation optimization framework. This framework is capable of reducing the computation time, improving the quality of optimal solutions, and increasing the confidence level in the optimality of the optimal solutions. This paper proposes the integration of common random numbers and parallel computing to achieve the stated objective. The parallel computing is performed on a single multicore processor. Based on the case study, the proposed framework was able to reduce the computation time by 90.5%, achieve a speedup of 2, improve the hypervolume indicator by 3.44%, and increase the confidence level by at least 100%. The values of improvement achieved will not necessarily be the same when different hardware, simulation models, simulation software, and optimization algorithms are used. The proposed framework allows project planners to obtain superior optimal solutions faster, which will make the use of stochastic simulation optimization more appealing.

A methodology for optimization of simulation models is presented. The methodology is based on a genetic algorithm in conjunction with an indifference-zone ranking and selection procedure under common random numbers. An application of this optimization algorithm to a stochastic mathematical model is provided.