Expensive Black-box Functions Research Articles

Deephive: A Reinforcement Learning Approach for Automated Discovery of Swarm-Based Optimization Policies

We present an approach for designing swarm-based optimizers for the global optimization of expensive black-box functions. In the proposed approach, the problem of finding efficient optimizers is framed as a reinforcement learning problem, where the goal is to find optimization policies that require a few function evaluations to converge to the global optimum. The state of each particle within the swarm is defined as its current position and function value within a design space, and the particles learn to take favorable actions that maximize the reward, which is based on the final value of the objective function. The proposed approach is tested on 50 benchmark optimization functions and compared to the performance of other global optimization strategies. Furthermore, the generalization capabilities of the trained particles on the four categories of optimization benchmark functions are investigated. The results show superior performance compared to the other optimizers, desired scaling when the dimension of the functions is varied, and acceptable performance even when applied to unseen functions. On a broader scale, the results show promise for the rapid development of domain-specific optimizers.

Solving higher dimensional expensive black box global optimization problems using sparse directional search on surrogates

We present a new algorithm Directional Optimization Search with Surrogate (DOSS), for optimizing problems with box constraints and a computationally expensive black-box objective function. DOSS is a radial basis function (RBF) based method that mainly focuses on higher dimensional and computationally expensive objective functions that can be multimodal. DOSS introduces three new techniques not previously used in earlier RBF algorithms, including using a combination of the coordinates knowledge level, fewer initial sampling points, and the surrogate’s gradient information. Numerical results on a test set including 14 test problems with 36, 48, and 60 dimensions show that DOSS outperforms two recently published algorithms RBFOpt and TuRBO, and earlier RBF algorithms such as DYCORS. TuRBO is a Gaussian process based optimization algorithm, which outperformed earlier state-of-the-art methods. DOSS algorithm also has a good performance on real-world optimization problems, including robot pushing and rover trajectory planning problems. Almost sure convergence for the DOSS algorithm is also proven in this paper. An implementation of DOSS is available at https://doi.org/10.5281/zenodo.13731558.

Constrained Bayesian optimization with a cardiovascular application

The present paper investigates constrained global optimization techniques for computationally expensive black box functions that are globally defined but subject to some a priori unknown taboo regions. This challenge typically arises in healthcare applications, where the goal is to maximize the efficacy of a drug while staying within critical safety limits imposed by an external medical regulator. Motivated by the additional challenge where the constrained global optimum lies along the constraint boundary, we comparatively assess the performance of established optimization methods coupled with different acquisition functions in terms of accuracy and efficiency on the physiological application aforementioned and several benchmark problems representative of the complexity of the physiological application. We find the best method based on an average score computed across all applications. We also propose an ensemble method combining results from individual methods, which vastly outperforms the best average method. Furthermore, our study provides a thorough qualitative analysis of the optimization results, emphasizing the challenges a user may encounter when applying Bayesian optimization on constrained optimization problems.

Efficient trust region filter modeling strategies for computationally expensive black-box optimization

The study and application of contemporary optimization techniques considerably enhance the efficiency of chemical research and manufacturing. With the dynamic progression of modern manufacturing technologies, the emergence of numerous black-box models characterized by inaccessible mathematical formulations and high evaluation costs poses new challenges to traditional optimization methods, leading to difficulty in programming and solving. Hence, based on the trust region filter (TRF) method, we define a new framework to elevate optimization efficiency in this study for chemical systems involving computationally expensive black-box functions. Sampling size per iteration is reduced, and sampling efficiency is improved by incorporating the known data beyond the trust region to assist in constructing reduced models, which is achieved by the application of the Gaussian process. Through comparison and validation of benchmark tests and case studies, this approach demonstrates that using the Gaussian process as the reduced model can lower the number of calls to black-box functions by more than half compared to common linear and quadratic models, and convergence to first-order critical points can still be guaranteed.

Managing low–acuity patients in an Emergency Department through simulation–based multiobjective optimization using a neural network metamodel

This paper deals with Emergency Department (ED) fast-tracks for low-acuity patients, a strategy often adopted to reduce ED overcrowding. We focus on optimizing resource allocation in minor injuries units, which are the ED units that can treat low-acuity patients, with the aim of minimizing patient waiting times and ED operating costs. We formulate this problem as a general multiobjective simulation-based optimization problem where some of the objectives are expensive black-box functions that can only be evaluated through a time-consuming simulation. To efficiently solve this problem, we propose a metamodeling approach that uses an artificial neural network to replace a black-box objective function with a suitable model. This approach allows us to obtain a set of Pareto optimal points for the multiobjective problem we consider, from which decision-makers can select the most appropriate solutions for different situations. We present the results of computational experiments conducted on a real case study involving the ED of a large hospital in Italy. The results show the reliability and effectiveness of our proposed approach, compared to the standard approach based on derivative-free optimization.

Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization

We consider the problem of multi-objective optimization (MOO) of expensive black-box functions with the goal of discovering high-quality and diverse Pareto fronts where we are allowed to evaluate a batch of inputs. This problem arises in many real-world applications including penicillin production where diversity of solutions is critical. We solve this problem in the framework of Bayesian optimization (BO) and propose a novel approach referred to as Pareto front-Diverse Batch Multi-Objective BO (PDBO). PDBO tackles two important challenges: 1) How to automatically select the best acquisition function in each BO iteration, and 2) How to select a diverse batch of inputs by considering multiple objectives. We propose principled solutions to address these two challenges. First, PDBO employs a multi-armed bandit approach to select one acquisition function from a given library. We solve a cheap MOO problem by assigning the selected acquisition function for each expensive objective function to obtain a candidate set of inputs for evaluation. Second, it utilizes Determinantal Point Processes (DPPs) to choose a Pareto-front-diverse batch of inputs for evaluation from the candidate set obtained from the first step. The key parameters for the methods behind these two steps are updated after each round of function evaluations. Experiments on multiple MOO benchmarks demonstrate that PDBO outperforms prior methods in terms of both the quality and diversity of Pareto solutions.

Stochastic Bayesian Optimization with Unknown Continuous Context Distribution via Kernel Density Estimation

Bayesian optimization (BO) is a sample-efficient method and has been widely used for optimizing expensive black-box functions. Recently, there has been a considerable interest in BO literature in optimizing functions that are affected by context variable in the environment, which is uncontrollable by decision makers. In this paper, we focus on the optimization of functions' expectations over continuous context variable, subject to an unknown distribution. To address this problem, we propose two algorithms that employ kernel density estimation to learn the probability density function (PDF) of continuous context variable online. The first algorithm is simpler, which directly optimizes the expectation under the estimated PDF. Considering that the estimated PDF may have high estimation error when the true distribution is complicated, we further propose the second algorithm that optimizes the distributionally robust objective. Theoretical results demonstrate that both algorithms have sub-linear Bayesian cumulative regret on the expectation objective. Furthermore, we conduct numerical experiments to empirically demonstrate the effectiveness of our algorithms.

D 3 PBO: D ynamic D omain D ecomposition-based P arallel B ayesian O ptimization for Large-scale Analog Circuit Sizing

Bayesian optimization (BO) is an efficient global optimization method for expensive black-box functions, but the expansion for high-dimensional problems and large sample budgets still remains a severe challenge. In order to extend BO for large-scale analog circuit synthesis, a novel computationally efficient parallel BO method, D 3 PBO, is proposed for high-dimensional problems in this work. We introduce the dynamic domain decomposition method based on maximum variance between clusters. The search space is decomposed into subdomains progressively to limit the maximal number of observations in each domain. The promising domain is explored by multi-trust region-based batch BO with the local Gaussian process (GP) model. As the domain decomposition progresses, the basin-shaped domain is identified using a GP-assisted quadratic regression method and exploited by the local search method BOBYQA to achieve a faster convergence rate. The time complexity of D 3 PBO is constant for each iteration. Experiments demonstrate that D 3 PBO obtains better results with significantly less runtime consumption compared to state-of-the-art methods. For the circuit optimization experiments, D 3 PBO achieves up to 10× runtime speedup compared to TuRBO with better solutions.

Lower Bounds on the Noiseless Worst-Case Complexity of Efficient Global Optimization

Efficient global optimization is a widely used method for optimizing expensive black-box functions. In this paper, we study the worst-case oracle complexity of the efficient global optimization problem. In contrast to existing kernel-specific results, we derive a unified lower bound for the oracle complexity of efficient global optimization in terms of the metric entropy of a ball in its corresponding reproducing kernel Hilbert space. Moreover, we show that this lower bound nearly matches the upper bound attained by non-adaptive search algorithms, for the commonly used squared exponential kernel and the Matérn kernel with a large smoothness parameter ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}. This matching is up to a replacement of d/2 by d and a logarithmic term logRϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log \\frac{R}{\\epsilon }$$\\end{document}, where d is the dimension of input space, R is the upper bound for the norm of the unknown black-box function, and ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} is the desired accuracy. That is to say, our lower bound is nearly optimal for these kernels.

Leveraging trust for joint multi-objective and multi-fidelity optimization

In the pursuit of efficient optimization of expensive-to-evaluate systems, this paper investigates a novel approach to Bayesian multi-objective and multi-fidelity (MOMF) optimization. Traditional optimization methods, while effective, often encounter prohibitively high costs in multi-dimensional optimizations of one or more objectives. Multi-fidelity approaches offer potential remedies by utilizing multiple, less costly information sources, such as low-resolution approximations in numerical simulations. However, integrating these two strategies presents a significant challenge. We propose the innovative use of a trust metric to facilitate the joint optimization of multiple objectives and data sources. Our methodology introduces a modified multi-objective (MO) optimization policy incorporating the trust gain per evaluation cost as one of the objectives of a Pareto optimization problem. This modification enables simultaneous MOMF optimization, which proves effective in establishing the Pareto set and front at a fraction of the cost. Two specific methods of MOMF optimization are presented and compared: a holistic approach selecting both the input parameters and the fidelity parameter jointly, and a sequential approach for benchmarking. Through benchmarks on synthetic test functions, our novel approach is shown to yield significant cost reductions—up to an order of magnitude compared to pure MO optimization. Furthermore, we find that joint optimization of the trust and objective domains outperforms sequentially addressing them. We validate our findings with the specific use case of optimizing particle-in-cell simulations of laser-plasma acceleration, highlighting the practical potential of our method in the Pareto optimization of highly expensive black-box functions. Implementation of the methods in existing Bayesian optimization frameworks is straightforward, with immediate extensions e.g. to batch optimization possible. Given their ability to handle various continuous or discrete fidelity dimensions, these techniques have wide-ranging applicability in tackling simulation challenges across various scientific computing fields such as plasma physics and fluid dynamics.

Constrained multiobjective optimization of expensive black-box functions using a heuristic branch-and-bound approach

Constrained multiobjective optimization of expensive black-box functions using a heuristic branch-and-bound approach

Adaptive surrogate assisted multi-objective optimization approach for highly nonlinear and complex engineering design problems

Despite enormous advances in computer power, computationally costly models impede the use of traditional optimization approaches that must be invoked repeatedly during the optimization process in practical engineering applications. Surrogate models have been found to be a promising endeavor in multi-objective optimization problems involving expensive analysis and simulation processes such as multi-physics modeling and simulation, finite element analysis (FEA), and computational fluid dynamics (CFD. Developing an optimization algorithm that can easily identify the Pareto frontier of highly nonlinear multi-objective optimization problems with less computation cost is the aim of this work. In this paper, an Adaptive Multi-Objective Optimization approach based Surrogate models (AMOS) is developed to reduce computation cost of fitness evaluations and discover the Pareto optima for multi-objective optimization problems with comparable high accuracy. AMOS explores the design space by sampling using LHD to identify promising regions. Then, AMOS exploits the identified promising region by adaptively constructing the most suitable surrogate model, which could be response surface, radial basis, or Kriging surrogates, in the feasible design space based on root mean square error values (RMSE). AMOS stops iterating when a termination criterion is met, and a Pareto frontier is identified based on developed guidance and fitness functions. AMOS has successfully identified the pareto frontier of practical engineering optimization problems with expensive black box functions and significantly reduced the computation cost. The novel method was put to the test utilizing real-world challenges and engineering design examples such vehicle magnetorheological design and wind turbine airfoil geometry.

Benchmarking surrogate-based optimisation algorithms on expensive black-box functions

Surrogate algorithms such as Bayesian optimisation are especially designed for black-box optimisation problems with expensive objectives, such as hyperparameter tuning or simulation-based optimisation. In the literature, these algorithms are usually evaluated with synthetic benchmarks which are well established but have no expensive objective, and only on one or two real-life applications which vary wildly between papers. There is a clear lack of standardisation when it comes to benchmarking surrogate algorithms on real-life, expensive, black-box objective functions. This makes it very difficult to draw conclusions on the effect of algorithmic contributions and to give substantial advice on which method to use when. A new benchmark library, EXPObench, provides first steps towards such a standardisation. The library is used to provide an extensive comparison of six different surrogate algorithms on four expensive optimisation problems from different real-life applications. This has led to new insights regarding the relative importance of exploration, the evaluation time of the objective, and the used model. We also provide rules of thumb for which surrogate algorithm to use in which situation. A further contribution is that we make the algorithms and benchmark problem instances publicly available, contributing to more uniform analysis of surrogate algorithms. Most importantly, we include the results of the six algorithms on all evaluated problem instances. This unique new dataset lowers the bar for researching new methods as the number of expensive evaluations required for comparison and for the creation of new surrogate models is significantly reduced.

Multi-Objective Bayesian Optimization Supported by an Expected Pareto Distance Change

Abstract The solution to global (a posteriori) multi-objective optimization problems traditionally relies on population-based algorithms, which are very effective in generating a Pareto front. Unfortunately, due to the high number of function evaluations, these methods are of limited use in problems that involve expensive black-box functions. In recent years, multi-objective Bayesian optimization has emerged as a powerful alternative; however, in many applications, these methods fail to generate a diverse and well-spread Pareto front. To address this limitation, our work introduces a novel acquisition function (AF) for multi-objective Bayesian optimization that produces more informative acquisition landscapes. The proposed AF comprises two terms, namely, a distance-based metric and a diversity index. The distance-based metric, referred to as the expected Pareto distance change, promotes the evaluation of high-performing designs and repels low-performing design zones. The diversity term prevents the evaluation of designs that are similar to the ones contained in the current sampling plan. The proposed AF is studied using seven analytical problems and in the design optimization of sandwich composite armors for blast mitigation, which involves expensive finite element simulations. The results show that the proposed AF generates high-quality Pareto sets outperforming well-established methods such as the Euclidean-based expected improvement function. The proposed AF is also compared with respect to a recently proposed multi-objective approach. The difference in their performance is problem dependent.

Parallel efficient global optimization by using the minimum energy criterion

In optimization problems, the expensive black-box function implies severely restricted budgets in terms of evaluation. Some Bayesian optimization methods have been proposed to solve this problem, such as expected improvement (EI) and hierarchical expected improvement (HEI). Neither EI nor HEI is parallel, which depends on a one-point-at-a-time strategy. In this work, a new parallel Bayesian framework based on the minimum energy criterion is proposed to improve these popular one-point methods. The new proposed framework can save time and costs by reducing the number of iterations and avoid the local optimization trap by encouraging the exploration of the optimization space. Additionally, a shrink-augment strategy is also introduced to correct the local surrogate model for the black-box function adaptively, which could also benefit the optimization. Some numerical and illustrative experiments are presented to demonstrate the superiority of our proposed method over some other Bayesian methods. The results show that the novel framework can balance exploitation and exploration well and has great performance in global optimization.

Ability of Black-Box Optimisation to Efficiently Perform Simulation Studies in Power Engineering

Abstract In this study, the potential of the so-called black-box optimisation (BBO) to increase the efficiency of simulation studies in power engineering is evaluated. Three algorithms (“Multilevel Coordinate Search” (MCS) and “Stable Noisy Optimization by Branch and Fit” (SNOBFIT) by Huyer and Neumaier and “blackbox: A Procedure for Parallel Optimization of Expensive Black-box Functions” (blackbox) by Knysh and Korkolis) are implemented in MATLAB and compared for solving two use cases: the analysis of the maximum rotational speed of a gas turbine after a load rejection and the identification of transfer function parameters by measurements. The first use case has a high computational cost, whereas the second use case is computationally cheap. For each run of the algorithms, the accuracy of the found solution and the number of simulations or function evaluations needed to determine the optimum and the overall runtime are used to identify the potential of the algorithms in comparison to currently used methods. All methods provide solutions for potential optima that are at least 99.8% accurate compared to the reference methods. The number of evaluations of the objective functions differs significantly but cannot be directly compared as only the SNOBFIT algorithm does stop when the found solution does not improve further, whereas the other algorithms use a predefined number of function evaluations. Therefore, SNOBFIT has the shortest runtime for both examples. For computationally expensive simulations, it is shown that parallelisation of the function evaluations (SNOBFIT and blackbox) and quantisation of the input variables (SNOBFIT) are essential for the algorithmic performance. For the gas turbine overspeed analysis, only SNOBFIT can compete with the reference procedure concerning the runtime. Further studies will have to investigate whether the quantisation of input variables can be applied to other algorithms and whether the BBO algorithms can outperform the reference methods for problems with a higher dimensionality.

Bayesian optimization for demographic inference.

Inference of demographic histories of species and populations is one of the central problems in population genetics. It is usually stated as an optimization problem: find a model's parameters that maximize a certain log-likelihood. This log-likelihood is often expensive to evaluate in terms of time and hardware resources, critically more so for larger population counts. Although genetic algorithm-based solution has proven efficient for demographic inference in the past, it struggles to deal with log-likelihoods in the setting of more than three populations. Different tools are therefore needed to handle such scenarios. We introduce a new optimization pipeline for demographic inference with time consuming log-likelihood evaluations. It is based on Bayesian optimization, a prominent technique for optimizing expensive black box functions. Comparing to the existing widely used genetic algorithm solution, we demonstrate new pipeline's superiority in the limited time budget setting with four and five populations, when using the log-likelihoods provided by the moments tool.

Representative points for distribution recovering

This paper studies a scheme of producing representative points to approximate an arbitrary distribution or the unknown distribution of a given data. It leverages on the rich literature of space-filling designs on the unit hypercube, picks a good design and transforms it toward the distribution. This new approach is flexible and accurate in estimating the expected value of an expensive blackbox function over the underlying distribution. It accommodates any dependence structure among variables, and allows for any shape of supporting regions, including non-convex ones. Nested and sliced structures of the space-filling designs on the unit hypercube can be seamlessly preserved in the transformed space. Asymptotic properties of the resulting estimators are also established.

A Nested Weighted Tchebycheff Multi-Objective Bayesian Optimization Approach for Flexibility of Unknown Utopia Estimation in Expensive Black-Box Design Problems

Abstract We propose a nested weighted Tchebycheff Multi-objective Bayesian optimization (WTB MOBO) framework where we built a regression model selection procedure from the ensemble of models, toward better estimation of the uncertain parameters (utopia) of the weighted Tchebycheff expensive black-box multi-objective function. In our previous work, a weighted Tchebycheff MOBO approach has been demonstrated which attempts to estimate the model parameters (utopia) in formulating the acquisition function of the weighted Tchebycheff multi-objective black-box functions, through calibration using an a priori selected regression model. However, the existing MOBO model lacks flexibility in selecting the appropriate regression models given the guided sampled data and, therefore, can under-fit or over-fit as the iterations of the MOBO progress. This ultimately can reduce the overall MOBO performance. As, in general, it is too complex to a priori guarantee a best model, this motivates us to consider a portfolio of different families (simple-to-complex) of predictive models that have been fitted with current training data guided by the WTB MOBO, and the best model is selected following a user-defined prediction root-mean-square error-based approach. The proposed approach is implemented in optimizing a thin tube design under constant loading of temperature and pressure, minimizing the risk of creep-fatigue failure and design cost. Finally, the nested WTB MOBO model performance is compared with different MOBO frameworks with respect to accuracy in parameter estimation, Pareto-optimal solutions, and function evaluation cost. This approach is generalized enough to consider different families of predictive models in the portfolio for best model selection, where the overall design architecture allows for solving any high-dimensional (multiple functions) complex black-box problems and can be extended to any other global criterion multi-objective optimization methods where prior knowledge of utopia is required.

Interval uncertainty propagation by a parallel Bayesian global optimization method

This paper is concerned with approximating the scalar response of a complex computational model subjected to multiple input interval variables. Such task is formulated as finding both the global minimum and maximum of a computationally expensive black-box function over a prescribed hyper-rectangle. On this basis, a novel non-intrusive method, called ‘triple-engine parallel Bayesian global optimization’, is proposed. The method begins by assuming a Gaussian process prior (which can also be interpreted as a surrogate model) over the response function. The main contribution lies in developing a novel infill sampling criterion, i.e., triple-engine pseudo expected improvement strategy, to identify multiple promising points for minimization and/or maximization based on the past observations at each iteration. By doing so, these identified points can be evaluated on the real response function in parallel. Besides, another potential benefit is that both the lower and upper bounds of the model response can be obtained with a single run of the developed method. Four numerical examples with varying complexity are investigated to demonstrate the proposed method against some existing techniques, and results indicate that significant computational savings can be achieved by making full use of prior knowledge and parallel computing.