Black-box Optimization Problems Research Articles

Identifying Parameters of Reactor Operation Digital Twin by Using Swarm and Evolutionary Algorithms

Digital twin (DT) technology is currently gaining popularity in a variety of industrial applications. Recently, the China Institute of Nuclear Power proposed a data driven by nuclear reactor digital twin. The digital twin solves the forward problem for physical field simulation and the inverse problem for parameter identification. The inverse problem can be viewed as a black-box optimization problem where the input parameters are identified by the Latin hypercube sampling (LHS) method based on a given partial observation. While LHS is relatively naive, in this paper, we proposed to apply swarm and evolutionary algorithms to solve the inverse problem. Specifically, we present an experimental comparative study for LHS and four popular SEAs, Particle Swarm Optimizer (PSO), Differential Evolution (DE), Evolutionary Strategy (ES), and evolution strategy with covariance matrix adaptation (CMA-ES). The results show that, for this inverse problem, SEAs are more accurate and efficient than LHS. We also investigate the comparative optimization algorithm’s searching capability under the conditions of noisy observations and no initial input parameter guesses. The experimental results show that DE and CMA-ES have strong robustness and high searching ability.

Retraction-Based Direct Search Methods for Derivative Free Riemannian Optimization

AbstractDirect search methods represent a robust and reliable class of algorithms for solving black-box optimization problems. In this paper, the application of those strategies is exported to Riemannian optimization, wherein minimization is to be performed with respect to variables restricted to lie on a manifold. More specifically, classic and linesearch extrapolated variants of direct search are considered, and tailored strategies are devised for the minimization of both smooth and nonsmooth functions, by making use of retractions. A class of direct search algorithms for minimizing nonsmooth objectives on a Riemannian manifold without having access to (sub)derivatives is analyzed for the first time in the literature. Along with convergence guarantees, a set of numerical performance illustrations on a standard set of problems is provided.

A Right-Hand Side Function Surrogate Model-Based Method for the Black-Box Dynamic Optimization Problem

Abstract When solving the black-box dynamic optimization problem (BDOP) in the sophisticated dynamic system, the finite difference technique requires significant computational efforts on numerous expensive system simulations to provide approximate numerical Jacobian information for the gradient-based optimizer. To save computational budget, this work introduces a BDOP solving framework based on the right-hand side (RHS) function surrogate model (RHSFSM), in which the RHS derivative functions of the state equation are approximated by the surrogate models, and the Jacobian information is provided by inexpensive estimations of RHSFSM rather than the original time-consuming system evaluations. Meanwhile, the sampling strategies applicable to the construction of RHSFSM are classified into three categories: direct, indirect, and hybrid sampling strategy, and the properties of these strategies are analyzed and compared. Furthermore, to assist the RHSFSM-based BDOP solving framework search for the optimum efficiently, a novel dynamic hybrid sampling strategy is proposed to update RHSFSM sequentially. Finally, two dynamic optimization examples and a co-design example of a horizontal axis wind turbine illustrate that the RHSFSM-based BDOP solving framework integrated with the proposed dynamic hybrid sampling strategy not only solves the BDOP efficiently but also achieves the optimal solution robustly and reliably compared to other sampling strategies.

Discrete simulation optimization for tuning machine learning method hyperparameters

ABSTRACT An important aspect of machine learning (ML) involves controlling the learning process for the ML method in question to maximize its performance. Hyperparameter tuning (HPT) involves selecting suitable ML method parameters that control its learning process. Given that HPT can be conceptualized as a black box optimization problem subject to stochasticity, simulation optimization (SO) methods appear well suited to this purpose. Therefore, we conceptualize HPT as a discrete SO problem and demonstrate the use of the Kim and Nelson (KN) ranking and selection method, and the stochastic ruler (SR) and the adaptive hyperbox (AH) random search methods for HPT. We also construct the theoretical basis for applying the KN method. We demonstrate the application of the KN and the SR methods to a wide variety of machine learning models, including deep neural network models. We then successfully benchmark the KN, SR and the AH methods against multiple state-of-the-art HPT methods.

Personalized Bayesian optimization for noisy problems

In many real-world applications of interest, several related optimization tasks can be encountered, where each task is associated with a specific context or personalized information. Moreover, the amount of available data for each task may be highly limited due to the expensive cost involved. Although Bayesian optimization (BO) has emerged as a promising paradigm for handling black-box optimization problems, addressing such a sequence of optimization tasks can be intractable due to the cold start issues in BO. The key challenge is to speed up the optimization by leveraging the transferable information, while taking the personalization into consideration. In this paper, optimization problems with personalized variables are formally defined at first. Subsequently, a personalized evolutionary Bayesian algorithm is proposed to consider the personalized information and the measurement noise. Specifically, a contextual Gaussian process is used to jointly learn a surrogate model in different contexts with regard to the varying personalized parameter, and an evolutionary algorithm is tailored for optimizing an acquisition function for handling the presence of personalized information. Finally, we demonstrate the effectiveness of the proposed algorithm by testing it on widely used single- and multi-objective benchmark problems with personalized variables.

Comparative analysis of machine learning methods for active flow control

Machine learning frameworks such as genetic programming and reinforcement learning (RL) are gaining popularity in flow control. This work presents a comparative analysis of the two, benchmarking some of their most representative algorithms against global optimization techniques such as Bayesian optimization and Lipschitz global optimization. First, we review the general framework of the model-free control problem, bringing together all methods as black-box optimization problems. Then, we test the control algorithms on three test cases. These are (1) the stabilization of a nonlinear dynamical system featuring frequency cross-talk, (2) the wave cancellation from a Burgers’ flow and (3) the drag reduction in a cylinder wake flow. We present a comprehensive comparison to illustrate their differences in exploration versus exploitation and their balance between ‘model capacity’ in the control law definition versus ‘required complexity’. Indeed, we discovered that previous RL control attempts of controlling the cylinder wake were performing linear control and that the wide observation space was limiting their performances. We believe that such a comparison paves the way towards the hybridization of the various methods, and we offer some perspective on their future development in the literature of flow control problems.

A General Mathematical Framework for Constrained Mixed-variable Blackbox Optimization Problems with Meta and Categorical Variables

A mathematical framework for modelling constrained mixed-variable optimization problems is presented in a blackbox optimization context. The framework introduces a new notation and allows solution strategies. The notation framework allows meta and categorical variables to be explicitly and efficiently modelled, which facilitates the solution of such problems. The new term meta variables is used to describe variables that influence which variables are included or excluded: meta variables may affect the number of variables and constraints. The flexibility of the solution strategies supports the main blackbox mixed-variable optimization approaches: direct search methods and surrogate-based methods (Bayesian optimization). The notation system and solution strategies are illustrated through an example of a hyperparameter optimization problem from the machine learning community.

Exploring parameter spaces with artificial intelligence and machine learning black-box optimization algorithms

Constraining beyond the Standard Model theories usually involves scanning highly multidimensional parameter spaces and checking observable predictions against experimental bounds and theoretical constraints. Such a task is often timely and computationally expensive, especially when the model is severely constrained and thus leading to very low random sampling efficiency. In this work we tackled this challenge using artificial intelligence and machine learning search algorithms used for black-box optimization problems. Using the constrained minimal supersymmetric standard model and the phenomenological minimal supersymmetric standard model parameter spaces, we consider both the Higgs mass and the dark matter relic density constraints to study their sampling efficiency and parameter space coverage. We find our methodology to produce orders of magnitude improvement of sampling efficiency while reasonably covering the parameter space.

An Evolutionary Guardrail Layout Design Framework for Crowd Control in Subway Stations

Deploying guardrails near elevator entrances is an effective way to alleviate congestion and improve the flow rate in subway stations. How to properly design the guardrail layout is a complex black-box optimization problem. Existing methods are mainly based on manual design, which are highly dependent on the empirical experience of the designers and may not get satisfactory results in complicated scenarios. To address the above issues, this article proposes an evolutionary framework to automatically optimize guardrail layouts in subway stations. In the proposed framework, a novel guardrail layout encoding method is proposed, which can facilitate the algorithm to generate regular guardrail layout design solutions. Furthermore, a new fitness evaluation function is proposed to effectively measure the quality of a given guardrail layout design strategy. To validate its effectiveness, the proposed framework is applied to two scenarios with different characteristics. Simulation results have demonstrated that the proposed framework can provide promising guardrail layout designs, which can alleviate the congestion of subway stations effectively.

Blackbox optimization and surrogate models for machining free-form surfaces

This paper introduces an optimization model for machining free-form surfaces. It involves one categorical decision variable and continuous decision variables. Its objective function is partially separable. It is composed of two blackboxes: a clustering blackbox that outputs a partition of the surface into K zones, and K independent blackboxes, each of which outputs a machining time for a zone. This blackbox optimization problem is solved with the state-of-the-art software NOMAD. In order to improve the performance of the optimization process, we propose several surrogates of the machining-time blackboxes. Some of these surrogates are simple numerical approximations of the machining time, while one proposed surrogate is analytical, cheap to evaluate and exact for zones that are rectangles. Numerical experiments on two benchmark test surfaces show that our methodology outperforms other approaches from the literature. Although performances are strongly dependent on the topology of the test surfaces, the gains in machining time can go up to 40%.

Sensor Rotational Measurement with Virtual Sensors Applying Pearson Generative Adversarial Imputation Nets

Recent advances in sensor technology have increased the ability of humans to measure a wide range of phenomena and events. Nevertheless, in some cases, due to a variety of limitations, only a few sensors can be deployed at a given site. Consequently, setting up enough sensors at the right places to provide uniform monitoring can be difficult. In addition, virtual sensing, which is a set of strategies for replacing a portion of physical sensors with virtual sensors, has recently been developed. Therefore, this work leverages the imputation capability of PGAIN-VS to develop a black-box data-driven virtual sensing approach named sensor rotational measurement for the purpose of reducing the number of physical devices to be used in reality while still ensuring monitoring accuracy. The approach takes advantage of the PGAIN-VS and Borda voting methods to determine the subset of real sensors that can take turns observing information within an interval of time. The approach is seen as a black-box objective optimization problem with constraints that is solved by the OpenBox tool. We evaluated our method on several real-world datasets and achieved promising results, with the overall number of physical sensors reduced by up to 20%.

Aberration correction for multiphoton microscopy using covariance matrix adaptation evolution strategy

Multiphoton microscopy is the enabling tool for biomedical research, but the aberrations of biological tissues have limited its imaging performance. Adaptive optics (AO) has been developed to partially overcome aberration to restore imaging performance. For indirect AO, algorithm is the key to its successful implementation. Here, based on the fact that indirect AO has an analogy to the black-box optimization problem, we successfully apply the covariance matrix adaptation evolution strategy (CMA-ES) used in the latter, to indirect AO in multiphoton microscopy (MPM). Compared with the traditional genetic algorithm (GA), our algorithm has a greater improvement in convergence speed and convergence accuracy, which provides the possibility of realizing real-time dynamic aberration correction for deep in vivo biological tissues.

Multi-agent Black-box Optimization using a Bayesian Approach to Alternating Direction Method of Multipliers*

Bayesian optimization (BO) is a powerful black-box optimization framework that looks to efficiently learn the global optimum of an unknown system by systematically trading-off between exploration and exploitation. However, the use of BO as a tool for coordinated decision-making in multi-agent systems with unknown structure has not been widely studied. This paper investigates a black-box optimization problem over a multi-agent network coupled via shared variables or constraints, where each subproblem is formulated as a BO that uses only its local data. The proposed multi-agent BO (MABO) framework adds a penalty term to traditional BO acquisition functions to account for coupling between the subsystems without data sharing. We derive a suitable form for this penalty term using alternating directions method of multipliers (ADMM), which enables the local decision-making problems to be solved in parallel (and potentially asynchronously). The effectiveness of the proposed MABO method is demonstrated on an intelligent transport system for fuel efficient vehicle platooning.

Adaptive Ranking-Based Constraint Handling for Explicitly Constrained Black-Box Optimization.

We propose a novel constraint-handling technique for the covariance matrix adaptation evolution strategy (CMA-ES). The proposed technique is aimed at solving explicitly constrained black-box continuous optimization problems, in which the explicit constraint is a constraint whereby the computational time for the constraint violation and its (numerical) gradient are negligible compared to that for the objective function. This method is designed to realize two invariance properties: invariance to the affine transformation of the search space, and invariance to the increasing transformation of the objective and constraint functions. The CMA-ES is designed to possess these properties for handling difficulties that appear in black-box optimization problems, such as non-separability, ill-conditioning, ruggedness, and the different orders of magnitude in the objective. The proposed constraint-handling technique (CHT), known as ARCH, modifies the underlying CMA-ES only in terms of the ranking of the candidate solutions. It employs a repair operator and an adaptive ranking aggregation strategy to compute the ranking. We developed test problems to evaluate the effects of the invariance properties, and performed experiments to empirically verify the invariance of the algorithm. We compared the proposed method with other CHTs on the CEC 2006 constrained optimization benchmark suite to demonstrate its efficacy. Empirical studies reveal that ARCH is able to exploit the explicitness of the constraint functions effectively, sometimes even more efficiently than an existing box-constraint handling technique on box-constrained problems, while exhibiting the invariance properties. Moreover, ARCH overwhelmingly outperforms CHTs by not exploiting the explicit constraints in terms of the number of objective function calls.

An Approximated Gradient Sign Method Using Differential Evolution for Black-Box Adversarial Attack

Recent studies show that deep neural networks are vulnerable to adversarial attacks in the form of subtle perturbations to the input image, which leads the model to output wrong prediction. Such an attack can easily succeed by the existing white-box attack methods, where the perturbation is calculated based on the gradient of the target network. Unfortunately, the gradient is often unavailable in the real-world scenarios, which makes the black-box adversarial attack problems practical and challenging. In fact, they can be formulated as high-dimensional black-box optimization problems at the pixel level. Although evolutionary algorithms are well known for solving black-box optimization problems, they cannot efficiently deal with the high-dimensional decision space. Therefore, we propose an approximated gradient sign method using differential evolution (DE) for solving black-box adversarial attack problems. Unlike most existing methods, it is novel that the proposed method searches the gradient sign rather than the perturbation by a DE algorithm. Also, we transform the pixel-based decision space into a dimension-reduced decision space by combining the pixel differences from the input image to neighbor images, and two different techniques for selecting neighbor images are introduced to build the transferred decision space. In addition, six variants of the proposed method are designed according to the different neighborhood selection and optimization search strategies. Finally, the performance of the proposed method is compared with a number of the state-of-the-art adversarial attack algorithms on CIFAR-10 and ImageNet datasets. The experimental results suggest that the proposed method shows superior performance for solving black-box adversarial attack problems, especially nontargeted attack problems.

A Generalized Framework of Multifidelity Max-Value Entropy Search Through Joint Entropy.

Bayesian optimization (BO) is a popular method for expensive black-box optimization problems; however, querying the objective function at every iteration can be a bottleneck that hinders efficient search capabilities. In this regard, multifidelity Bayesian optimization (MFBO) aims to accelerate BO by incorporating lower-fidelity observations available with a lower sampling cost. In our previous work, we proposed an information-theoretic approach to MFBO, referred to as multifidelity max-value entropy search (MF-MES), which inherits practical effectiveness and computational simplicity of the well-known max-value entropy search (MES) for the single-fidelity BO. However, the applicability of MF-MES is still limited to the case that a single observation is sequentially obtained. In this letter, we generalize MF-MES so that information gain can be evaluated even when multiple observations are simultaneously obtained. This generalization enables MF-MES to address two practical problem settings: synchronous parallelization and trace-aware querying. We show that the acquisition functions for these extensions inherit the simplicity of MF-MES without introducing additional assumptions. We also provide computational techniques for entropy evaluation and posterior sampling in the acquisition functions, which can be commonly used for all variants of MF-MES. The effectiveness of MF-MES is demonstrated using benchmark functions and real-world applications such as materials science data and hyperparameter tuning of machine-learning algorithms.

A Novel Surrogate Model-Based Solving Framework for the Black-Box Dynamic Co-Design and Optimization Problem in the Dynamic System

When encountering the black-box dynamic co-design and optimization (BDCDO) problem in the multidisciplinary dynamic system, the finite difference technique is inefficient or even infeasible to provide approximate numerical gradient information for the optimization algorithm since it requires numerous original expensive evaluations. Therefore, a solving framework based on the surrogate model of the state equation is introduced to optimize BDCDO. To efficiently construct the surrogate model, a sequential sampling method is presented on the basis of the successive relative improvement ratio. Meanwhile, a termination criterion is suggested to quantify the convergence of the solution. Ultimately, the newly proposed sampling strategy and termination criterion are incorporated into the BDCDO solving framework to optimize two numerical examples and two engineering examples. The results demonstrate that the framework integrating the proposed sampling strategy and termination criterion has the best performance in terms of the accuracy, efficiency, and computational budget compared to the existing methods.

An Improved Blind Kriging Surrogate Model for Design Optimization Problems

Surrogate modeling techniques are widely employed in solving constrained expensive black-box optimization problems. Therein, Kriging is among the most popular surrogates in which the trend function is considered as a constant mean. However, it also encounters several challenges related to capturing the overall trend with a relatively limited number of function evaluations as well as searching feasible points with complex or discontinuous feasible regions. To address this above issue, this paper presents an improved surrogate blind Kriging (IBK) and a combined infill strategy to find the optimal solution. According to enhancing the prediction accuracy of metamodels of objective and constraints, the high-order effects of regression function in the blind Kriging are identified by promising a variable selection technique. In addition, an infill strategy is developed based on the probability of feasibility, penalization, and constrained expected improvement for updating blind Kriging metamodels of the objective and constraints. At each iteration, two infill sample points are allocated at the positions to achieve improvement in optimality and feasibility. The IBK metamodels are updated by the newly-added infill sample points, which leads the proposed framework search to rapidly converge to the optimal solution. The performance and applicability of the proposed model are tested on several numerical benchmark problems via comparing with other metamodel-based constrained optimization methods. The obtained results indicate that IBK generally has a greater efficiency performance and outperforms the competitors in terms of a limited number of function evaluations. Finally, IBK is successfully applied to structural design optimization. The optimization results show that IBK is able to find the best feasible design with fewer evaluation functions compared with other studies, and this demonstrates the effectiveness and practicality of the proposed model for solving the constrained expensive black-box engineering design optimization problems.

Escaping Unknown Discontinuous Regions in Blackbox Optimization

The design of key nonlinear systems often requires the use of expensive blackbox simulations presenting inherent discontinuities whose positions in the variable space cannot be analytically predicted. Without further precautions, the solution of related optimization problems leads to design configurations which may be close to discontinuities of the blackbox output functions. These discontinuities may betray unsafe regions of the design space, such as nonlinear resonance regions. To account for possible changes of operating conditions, an acceptable solution must be away from unsafe regions of the space of variables. The objective of this work is to solve a constrained blackbox optimization problem with the additional constraint that the solution should be outside unknown zones of discontinuities or strong variations of the objective function or the constraints. The proposed approach is an extension of the mesh adaptive direct search (\sf Mads) algorithm and aims at building a series of inner approximations of these zones. The algorithm, called \sf DiscoMADS, relies on two main mechanisms: revealing discontinuities and progressively escaping the surrounding zones. A convergence analysis supports the algorithm and preserves the optimality conditions of \sf Mads. Numerical tests are conducted on analytical problems and on three engineering problems illustrating the following possible applications of the algorithm: the design of a simplified truss, the synthesis of a chemical component, and the design of a turbomachine blade. The \sf DiscoMADS algorithm successfully solves these problems by providing a feasible solution away from discontinuous regions.

An efficient global optimization algorithm for expensive constrained black-box problems by reducing candidate infilling region

Surrogate-based optimization methods are usually employed to solve expensive black-box optimization problems. Constraints bring more challenges for optimization than unconstrained problems. This study proposes a novel three-stage infilling framework to solve expensive single-objective problems with equality and/or inequality constraints. In the first stage, a criterion for locating a feasible region is designed. Until a feasible solution is found, the first stage is removed in the subsequent process. The second stage exploits the current best feasible solution and explores potentially better solutions. If the infilled site is infeasible, the third stage will proceed to improve the surrogate accuracy around the feasible region boundary and an extra constraint derived from the objective is proposed to reduce the candidate infill region in this stage. Otherwise, the process stays in stage two. Stage two continues once one third-stage point has been sampled. For brevity, the three-stage framework is called CEGO-DI. The efficacy of CEGO-DI is demonstrated by comparing it with two variants of it. In addition, the method is also compared with several state-of-the-art algorithms on extensive problems. The experimental results show that CEGO-DI has a better or competitive performance on most problems.