Black-box Optimization Algorithms Research Articles

Demonstration Learning and Generalization of Robotic Motor Skills Based on Wearable Motion Tracking Sensors

Learning from Demonstration (LfD) is an important way for robots to learn new skills. Its principle is to obtain an optimized trajectory through an instructor’s action teaching or action coding regression. By learning the teaching trajectory, the robot can master the movement skills. However, adapting the learned motor skills to meet additional constraints that arise during the task can be challenging. This paper proposed a noise exploration method PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BB</sup> -CMA, which improves the Policy Improvement through Black-Box Optimization (PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BB</sup> ) algorithm by using an adaptive covariance matrix. In our method, the teaching trajectory is collected with our designed inertial motion capture system. After then, a more reasonable trajectory is synthesized as the input to Dynamic Movement Primitives (DMP), using the Gaussian Mixture Model (GMM) and Gaussian Mixture Regression (GMR). Finally, the improved algorithm PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BB</sup> -CMA is used to perform noise exploration to ensure that the robot arm can pass the designated points in the path planning. Furthermore, three baseline methods are chosen for comparison to validate the robustness of our proposed method. The experimental results demonstrated that our proposed method outperforms the baseline algorithms and is feasible for improving robot intelligence. We believe that our method has a certain potential in the aspect of skill learning for robots.

Black-box optimization algorithms for chemical synthesis and reaction condition optimization

<p indent="0mm">With the development of automated experimental platform technology, self-optimization technology and reaction condition optimization have become a promising trend in chemical synthesis. Self-optimization techniques optimize reaction objective functions, such as yield, by adjusting various reaction condition parameters. Among self-optimization algorithms, black-box optimization algorithms are widely used in the optimization of reaction conditions in synthetic chemistry. This article outlines the application of black-box optimization algorithms in reaction condition optimization, and systematically introduces the encoding of chemical reaction, various black-box optimization algorithms and the existing public data and software.

Practical initialization of the Nelder–Mead method for computationally expensive optimization problems

Black-box optimization (BBO) algorithms are widely employed by practitioners to address computationally expensive real-world problems such as automatic tuning of machine learning models and evacuation route planning. The Nelder–Mead (NM) method is a well-known local search heuristic for BBO that has been applied to solve many real-world problems from way back because of its promising performance. However, this method has a strong dependence on initialization due to its local search tendency. Nevertheless, a discussion on the proper initialization of the NM method is limited to the recent study by Wessing (Optim Lett 13(4):847–856, 2019), which is solely based on an analysis using the simple sphere function. In this study, we take a further step to improve Wessing’s result by massively investigating how the initialization affects the search performance in views of the initial simplex size and shape and a constraint handling method that is employed on 24 BBO benchmarking problems. Based on the numerical results, we present the empirical best practice for the initialization of the NM method for cases involving a limited evaluation budget.

A Generator of Multiextremal Test Classes With Known Solutions for Black-Box-Constrained Global Optimization

A generator of classes of multidimensional test problems for benchmarking continuous constrained global optimization methods is proposed. It is based on the generator of test classes for global optimization proposed in 2003 by Gaviano, Kvasov, Lera, and Sergeyev and extends the previous generation procedure from the box-constrained case to the case of nonlinear constraints. The user has the possibility to fix the difficulty of tests in an intuitive way by choosing several types of constraints. A detailed information (including the global solution) for each of 100 problems in each generated class is provided to the user. The generator is particularly suited for testing black-box optimization algorithms that normally address low or medium dimensional problems with hard to evaluate objective functions.

Surrogate-based branch-and-bound algorithms for simulation-based black-box optimization

Surrogate-based branch-and-bound algorithms for simulation-based black-box optimization

Noisy Optimization by Evolution Strategies With Online Population Size Learning

Optimization modeling of real-world application problems usually involves noise from various sources. Noisy optimization imposes challenges to optimization methods since the objective values can be different for multiple evaluations. In this article, we propose a novel online population size learning (OPL) technique of evolution strategies for handling noisy optimization problems. By re-evaluating a fraction of the candidates, we measure the strength of noise level of the re-evaluated candidate solutions and adapt the population size according to the noise level. The proposed OPL combines the advantages of both explicit averaging by re-evaluations and the implicit averaging by large population size and overcomes their limitations. We incorporate it with the covariance matrix adaptation evolution strategy (CMA-ES) and obtain OPL-CMA-ES. Compared with the existing noise handling technique, the proposed OPL is much simpler in both concepts and computation. We conduct comprehensive experiments to evaluate the algorithm’s performance on standard problems with Gaussian noise. We further evaluate the performance of OPL-CMA-ES on the black-box optimization benchmarks (BBOBs) noisy testbed, which is a standard platform for comparing black-box optimization algorithms, compared with the state-of-the-art noise-handling algorithms. The experimental results show that OPL-CMA-ES achieves remarkable performance and outperforms the compared variants.

Advanced Simulation-Inserted Optimization Using Combined Quasi-Newton Method With Lagrangian Method for EM-Based Design Optimization

Traditional electromagnetic (EM)-based design optimizations typically treat the EM simulation as a black-box and use optimization algorithms to externally drive EM simulations iteratively. Recently, a novel EM-based design optimization technique, named as simulation-inserted optimization (SIO) in this article, has been developed by opening the black-box and inserting the optimization algorithm into the internal of EM simulation to achieve EM field solutions and the optimal values of design variables simultaneously. This article proposes an advanced SIO algorithm using combined quasi-Newton method with Lagrangian method. In the proposed algorithm, the finite element method (FEM) is used as the EM simulation method, and the Lagrangian method is incorporated to internally integrate EM simulation with design optimization. New formulations based on quasi-Newton method are derived to approximate the computationally intensive lower upper (LU) decomposition of the FEM system matrix in the current optimization iteration by that in the previous optimization iteration to significantly reduce the number of LU decompositions. The proposed SIO technique further derives novel optimization update formulations by introducing the line search algorithm to increase the robustness of optimization. Using the proposed SIO algorithm, only a few initial LU decompositions of FEM system matrices need to be calculated instead of repetitively calculating a large number of LU decompositions of FEM system matrices during optimization, consequently speeding up the overall optimization process. The proposed technique is demonstrated by two application examples of EM-based design optimization of microwave components.

A comprehensive investigation on novel center-based sampling for large-scale global optimization

During the last decade, metaheuristic algorithms have been well-established approaches which are utilized for solving complex real-world optimization problems. The most metaheuristic algorithms uses stochastic strategies in their initialization phase as well as during their new candidate generation steps when there is no a-priori knowledge about the solution, which is a common valid assumption for any black-box optimization algorithm. In recent years, researchers have introduced a new concept called center-based sampling which can be utilized in any search component of the optimization process, but so far it mainly has been used just for population initialization. This novel concept clarifies that in a search space, the center point has a higher chance to be closer to an unknown solution compared to a random point, especially when the dimension of the search space increases. Thus, this concept helps the optimizer to find a better solution in a shorter time. In this paper, a comprehensive study has been conducted on the effect of center-based sampling to solve an optimization problem using three different levels of detailed investigation. These levels are as the follows: 1) considering no specific algorithm and no specific landscape (i.e., Monte-Carlo simulation); 2) considering a specific landscape but no specific algorithm (i.e., random search vs. center-based random search), and finally, 3) considering a specific algorithm and specific landscape which includes the proposing three different schemes for using center-based sampling scheme for solving Large-scale Global Optimization (LSGO) problems effectively. Furthermore, in this study, we seek to investigate the properties and capabilities of center-based sampling during optimization, which can be extended to utilize it in machine learning too, because optimization is a key role player in search and learning models. The proposed methods in this paper are evaluated on CEC 2013 LSGO benchmark functions and a real-world optimization problem, i.e., evolving ANN on two medical data sets. The experimental results confirm that center-based sampling has a crucial impact in improving the convergence rate of optimization/search algorithms when solving high-dimensional optimization problems.

Deep models of superficial face judgments

The diversity of human faces and the contexts in which they appear gives rise to an expansive stimulus space over which people infer psychological traits (e.g., trustworthiness or alertness) and other attributes (e.g., age or adiposity). Machine learning methods, in particular deep neural networks, provide expressive feature representations of face stimuli, but the correspondence between these representations and various human attribute inferences is difficult to determine because the former are high-dimensional vectors produced via black-box optimization algorithms. Here we combine deep generative image models with over 1 million judgments to model inferences of more than 30 attributes over a comprehensive latent face space. The predictive accuracy of our model approaches human interrater reliability, which simulations suggest would not have been possible with fewer faces, fewer judgments, or lower-dimensional feature representations. Our model can be used to predict and manipulate inferences with respect to arbitrary face photographs or to generate synthetic photorealistic face stimuli that evoke impressions tuned along the modeled attributes.

Continuous black-box optimization with an Ising machine and random subspace coding

A black-box optimization algorithm such as Bayesian optimization finds the extremum of an unknown function by alternating the inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply Ising machines to overcome the difficulty in the continuous black-box optimization. As an Ising machine specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution by Ising machines has to be translated back. Our method has the following three parts: (1) random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector, (2) a quadratic unconstrained binary optimization (QUBO) defined by the acquisition function based on the nonnegative-weighted linear regression model, which is solved by Ising machines, and (3) a penalization scheme to ensure that the solution can be translated back. It is shown with benchmark tests that its performance using the D-Wave Advantage quantum annealer and simulated annealing is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up the possibility of Ising machines and other QUBO solvers, including a quantum approximate optimization algorithm using gated-quantum computers, and may expand its range of application to continuous-valued problems.

SpartaPlex: A deterministic algorithm with linear scalability for massively parallel global optimization of very large-scale problems

SpartaPlex is a novel black-box optimization algorithm that yields superior results to state-of-the-art optimizers under tight function evaluation budgets. SpartaPlex is compared with 11 state-of-the-art optimization algorithms on 24 n-dimensional and 6 application-based benchmark problems including mechanical, structural, and antenna design optimization. Using identical computing resources, SpartaPlex finds superior objective function solutions and, as dimensionality increases, executes at least an order of magnitude faster than prior state-of-the-art. SpartaPlex is additionally evaluated for scalability using a compute cluster with 46,080 cores across 16 Graphical Processing Units. Tested in up to 100,000 dimensions, SpartaPlex demonstrates linear scalability in time with respect to dimensionality and computational resources.

A novel inverse methodology for the extraction of bulk elasto-plastic tensile properties of metals using spherical instrumented indentation

This paper presents an inverse methodology relying on finite element simulation based optimization to extract material constitutive parameters from a single spherical indentation, which are used to estimate its bulk true stress–true strain curve. Methods of this type usually assume that the tensile curve of the studied material can be modelled by a power-law involving, for example, Young’s modulus, yield stress and a hardening coefficient. Since the tensile behaviour of some materials cannot be adequately fitted by such models, our approach rather relies on an optimization procedure which extracts a group of six points on the true stress–true strain curve, distributed as per a geometric progression in the strain space, as well as the elastic modulus. A direct search black-box optimization algorithm is used and is shown to be capable of eluding local minima. A surrogate step is introduced in the methodology, which is a simplified version of the inverse problem, to find a suitable starting point. The performance of the method is first investigated in a numerical study in which the estimated true stress–true strain curves lie within a maximum error of 5.7% from the corresponding target tensile curves, for four materials with different hardening behaviours. The maximum errors on the extracted elastic modulus and yield stress are 0.5% and 11.1%, respectively. A study of the convergence behaviour of the proposed method demonstrates an expected significant increase in computational times as compared to when using a hardening model. Finally, an experimental application of the method to two materials with a plastic plateau leads to true stress–true strain curves estimations with average differences of 3.9% and 0.9% with the macroscopic experimental tensile curve, respectively. The method is thus found to be accurate even in the presence of modelling and experimental errors.

A competitive variable-fidelity surrogate-assisted CMA-ES algorithm using data mining techniques

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm is one of the most successful continuous black-box optimization algorithms widely used in many fields. However, the algorithm would be computationally unaffordable due to the sharp increase in dimension and complexity of problems. To improve the computational efficiency and global optimizing ability, a competitive variable-fidelity surrogate-assisted CMA-ES (CVFS-CMA-ES) algorithm using data mining techniques is proposed in this paper. In the first data mining of CVFS-CMA-ES, the establishment method of the competitive variable-fidelity model (VFM) based on the fuzzy clustering algorithm is developed, aiming at making full use of high-fidelity information and focusing more on potential areas of design space. In the second data mining of CVFS-CMA-ES, the Lower Confidence Bound (LCB) method is modified by introducing the step size of CMA-ES to control the uncertain term adaptively, and the sample points are screened in each generation of CMA-ES based on the modified LCB method, which can make a balance between exploration and exploitation in the optimization process. The performance of CVFS-CMA-ES is compared with four known black-box optimization algorithms. Firstly, ten numerical examples of 10-dimensional and 30-dimensional benchmark functions are carried out, respectively. Moreover, a 20-dimensional engineering example of the aerospace variable-stiffness composite shell under combined loadings is studied in detail. Results of benchmark functions and engineering examples verify the high efficiency, robustness and high global optimizing ability of the proposed CVFS-CMA-ES in comparison to other algorithms.

Optimal Controller Design for Transient Stability Enhancement of Grid-Following Converters Under Weak-Grid Conditions

Modeling and design-oriented control of transient stability of grid-following converters have attained an increasing interest in recent years. Despite novel nonlinear models enabling a design-oriented enhanced transient stability controller, the focus has so far been limited to study only the synchronization dynamics of the phase-locked loop. To expand upon the knowledge of the large-signal performance and stability, this article proposes a systematic analysis procedure of a grid-following converter under weak-grid conditions and large-signal disturbances including the outer dc-link and ac-side voltage control loops. A reduced-order large-signal model is used to analyze the large-signal nonlinear behavior of the system using the area of the basin of attraction as a measure for large-signal robustness. Here, stabilizing and destabilizing trends for outer-loop controller parameters are given. Through a surrogate-model expensive black-box optimization algorithm, a computational-efficient optimal design of the outer-loop controller parameters is proposed to maximize the large-signal robustness. Finally, a recommendation and a design guideline for converter constraints and outer-loop controller parameters are given. This can be used to identify the influencing parameters for grid-following converters under large-signal disturbances, and as a tool for fast controller optimization toward large-signal robustness.

Black-Box Function Aerodynamic Topology Optimization Algorithm via Machine Learning Technologies

The objective of this research is to construct an efficient global topology optimization method using machine learning technologies. In the conventional design process of mechanical design, the conceptual design is the earliest stage of the design process; and it is carried out based on the designer’s own idea/experience. Therefore, it is difficult to obtain innovative and high-quality concepts overwhelming the designer’s knowledge since the earliest stage of the design process has the largest impact. In this research, therefore, the black-box function aerodynamic topology optimization algorithm via machine learning technologies (FANTOM) is developed to overcome the problem. In the FANTOM approach, topology optimization problems are solved using/combining two efficient global optimization methods developed by the authors: the efficient global optimization method for discontinuous optimization problems with infeasible regions using classification method, and the efficient global optimization method via clustering/classification methods and exploration strategy. In the present approach, topological optimal designs can be obtained only by setting an objective function and constraint conditions. The validity of the FANTOM approach is demonstrated in an inviscid drag minimization problem at a two-dimensional supersonic flow condition, which provides an optimal topology as the Busemann biplane airfoil. Executing topology optimizations with the variation in freestream Mach number, it is also demonstrated that the FANTOM approach can explore topological optimal designs robustly.

Natural evolutionary strategies for variational quantum computation

Natural evolutionary strategies (NES) are a family of gradient-free black-box optimization algorithms. This study illustrates their use for the optimization of randomly initialized parameterized quantum circuits (PQCs) in the region of vanishing gradients. We show that using the NES gradient estimator the exponential decrease in variance can be alleviated. We implement two specific approaches, the exponential and separable NES, for parameter optimization of PQCs and compare them against standard gradient descent. We apply them to two different problems of ground state energy estimation using variational quantum eigensolver and state preparation with circuits of varying depth and length. We also introduce batch optimization for circuits with larger depth to extend the use of ES to a larger number of parameters. We achieve accuracy comparable to state-of-the-art optimization techniques in all the above cases with a lower number of circuit evaluations. Our empirical results indicate that one can use NES as a hybrid tool in tandem with other gradient-based methods for optimization of deep quantum circuits in regions with vanishing gradients.

Analysis of Optimization Algorithms via Sum-of-Squares

We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms renders them particularly relevant for applications in machine learning and large-scale data analysis. Relying on sum-of-squares (SOS) optimization, we introduce a hierarchy of semidefinite programs that give increasingly better convergence bounds for higher levels of the hierarchy. Alluding to the power of the SOS hierarchy, we show that the (dual of the) first level corresponds to the performance estimation problem (PEP) introduced by Drori and Teboulle (Math Program 145(1):451–482, 2014), a powerful framework for determining convergence rates of first-order optimization algorithms. Consequently, many results obtained within the PEP framework can be reinterpreted as degree-1 SOS proofs, and thus, the SOS framework provides a promising new approach for certifying improved rates of convergence by means of higher-order SOS certificates. To determine analytical rate bounds, in this work, we use the first level of the SOS hierarchy and derive new results for noisy gradient descent with inexact line search methods (Armijo, Wolfe, and Goldstein).

DMulti-MADS: mesh adaptive direct multisearch for bound-constrained blackbox multiobjective optimization

The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available as the outputs of a blackbox for which no derivative information is available. This work proposes a new extension of the mesh adaptive direct search (MADS) algorithm to multiobjective derivative-free optimization with bound constraints. This method does not aggregate objectives and keeps a list of non dominated points which converges to a (local) Pareto set as long as the algorithm unfolds. As in the single-objective optimization MADS algorithm, this method is built around a search step and a poll step. Under classical direct search assumptions, it is proved that the so-called DMulti-MADS algorithm generates multiple subsequences of iterates which converge to a set of local Pareto stationary points. Finally, computational experiments suggest that this approach is competitive compared to the state-of-the-art algorithms for multiobjective blackbox optimization.

Black-Box Optimization for Automated Discovery.

In chemistry and materials science, researchers and engineers discover, design, and optimize chemical compounds or materials with their professional knowledge and techniques. At the highest level of abstraction, this process is formulated as black-box optimization. For instance, the trial-and-error process of synthesizing various molecules for better material properties can be regarded as optimizing a black-box function describing the relation between a chemical formula and its properties. Various black-box optimization algorithms have been developed in the machine learning and statistics communities. Recently, a number of researchers have reported successful applications of such algorithms to chemistry. They include the design of photofunctional molecules and medical drugs, optimization of thermal emission materials and high Li-ion conductive solid electrolytes, and discovery of a new phase in inorganic thin films for solar cells.There are a wide variety of algorithms available for black-box optimization, such as Bayesian optimization, reinforcement learning, and active learning. Practitioners need to select an appropriate algorithm or, in some cases, develop novel algorithms to meet their demands. It is also necessary to determine how to best combine machine learning techniques with quantum mechanics- and molecular mechanics-based simulations, and experiments. In this Account, we give an overview of recent studies regarding automated discovery, design, and optimization based on black-box optimization. The Account covers the following algorithms: Bayesian optimization to optimize the chemical or physical properties, an optimization method using a quantum annealer, best-arm identification, gray-box optimization, and reinforcement learning. In addition, we introduce active learning and boundless objective-free exploration, which may not fall into the category of black-box optimization.Data quality and quantity are key for the success of these automated discovery techniques. As laboratory automation and robotics are put forward, automated discovery algorithms would be able to match human performance at least in some domains in the near future.

Non-intrusive Time-POD for Optimal Control of a Fixed-Bed Reactor for CO2 Methanation

Abstract The optimization of a controlled process in a simulation without access to the model itself is a common scenario and very relevant to many chemical engineering applications. A general approach is to apply a black-box optimization algorithm to a parameterized control scheme. The success then depends on the quality of the parametrization that should be low-dimensional though rich enough to express the salient features. This work proposes using solution snapshots to extract dominant modes of the temporal dynamics of a process and use them for low-dimensional parametrizations of control functions. The approach is called proper orthogonal decomposition in time (time-POD). We provide theoretical reasoning and illustrate the performance for the optimal control of a methanation reactor.