Black-box Optimization Problems Research Articles

Multi-task collaborative method based on manifold optimization for automated test case generation based on path coverage

Automated test case generation based on path coverage is not only a large-scale black-box optimization problem, but also a key scientific problem in software automatic testing technology. Evolutionary algorithms and other search-based algorithms are representative methods for this problem. However, existing research mainly focuses on the multi-function case, where generating test cases for multiple functions is difficult due to the combinatorial explosion of the path number. In this paper, we propose a multi-task collaborative method based on manifold optimization by considering the topological manifold relationship between the test case space (decision space) and the program path space (target space). This method achieves the goal of collaborative optimization for different function coverage tasks by coordinating the allocation of computing resources and knowledge transfer mechanisms among them. To verify the effectiveness of the proposed method, we compare it with general solution methods for single-function automated test case generation based on path coverage, such as the manifold-inspired search-based algorithm. The experimental results show that the proposed method outperforms the compared single-function optimization algorithms especially on programs with strong coding similarities. This study verifies the feasibility of collaborative optimization methods for solving large-scale black-box optimization problems, such as the automated test case generation based on path coverage, and expands their application scenarios.

Communication-Efficient Zeroth-Order Adaptive Optimization for Federated Learning

Federated learning has become a prevalent distributed training paradigm, in which local devices collaboratively train learning models without exchanging local data. One of the most dominant frameworks of federated learning (FL) is FedAvg, since it is efficient and simple to implement; here, the first-order information is generally utilized to train the parameters of learning models. In practice, however, the gradient information may be unavailable or infeasible in some applications, such as federated black-box optimization problems. To solve the issue, we propose an innovative zeroth-order adaptive federated learning algorithm without using the gradient information, referred to as ZO-AdaFL, which integrates the zeroth-order optimization algorithm into the adaptive gradient method. Moreover, we also rigorously analyze the convergence behavior of ZO-AdaFL in a non-convex setting, i.e., where ZO-AdaFL achieves convergence to a region close to a stationary point at a speed of O(1/T) (T represents the total iteration number). Finally, to verify the performance of ZO-AdaFL, simulation experiments are performed using the MNIST and FMNIST datasets. Our experimental findings demonstrate that ZO-AdaFL outperforms other state-of-the-art zeroth-order FL approaches in terms of both effectiveness and efficiency.

Sample-Constrained Black Box Optimization for Audio Personalization

We consider the problem of personalizing audio to maximize user experience. Briefly, we aim to find a filter h*, which applied to any music or speech, will maximize the user’s satisfaction. This is a black-box optimization problem since the user’s satisfaction function is unknown. Substantive work has been done on this topic where the key idea is to play audio samples to the user, each shaped by a different filter hi, and query the user for their satisfaction scores f(hi). A family of “surrogate” functions is then designed to fit these scores and the optimization method gradually refines these functions to arrive at the filter ˆh* that maximizes satisfaction. In certain applications, we observe that a second type of querying is possible where users can tell us the individual elements h*[j] of the optimal filter h*. Consider an analogy from cooking where the goal is to cook a recipe that maximizes user satisfaction. A user can be asked to score various cooked recipes (e.g., tofu fried rice) or to score individual ingredients (say, salt, sugar, rice, chicken, etc.). Given a budget of B queries, where a query can be of either type, our goal is to find the recipe that will maximize this user’s satisfaction. Our proposal builds on Sparse Gaussian Process Regression (GPR) and shows how a hybrid approach can outperform any one type of querying. Our results are validated through simulations and real world experiments, where volunteers gave feedback on music/speech audio and were able to achieve high satisfaction levels. We believe this idea of hybrid querying opens new problems in black-box optimization and solutions can benefit other applications beyond audio personalization.

GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations

Traditional machine learning is generally treated as a black-box optimization problem and does not typically produce interpretable functions that connect inputs and outputs. However, the ability to discover such interpretable functions is desirable. In this work, we propose GINN-LP, an interpretable neural network to discover the form and coefficients of the underlying equation of a dataset, when the equation is assumed to take the form of a multivariate Laurent Polynomial. This is facilitated by a new type of interpretable neural network block, named the “power-term approximator block”, consisting of logarithmic and exponential activation functions. GINN-LP is end-to-end differentiable, making it possible to use backpropagation for training. We propose a neural network growth strategy that will enable finding the suitable number of terms in the Laurent polynomial that represents the data, along with sparsity regularization to promote the discovery of concise equations. To the best of our knowledge, this is the first model that can discover arbitrary multivariate Laurent polynomial terms without any prior information on the order. Our approach is first evaluated on a subset of data used in SRBench, a benchmark for symbolic regression. We first show that GINN-LP outperforms the state-of-the-art symbolic regression methods on datasets generated using 48 real-world equations in the form of multivariate Laurent polynomials. Next, we propose an ensemble method that combines our method with a high-performing symbolic regression method, enabling us to discover non-Laurent polynomial equations. We achieve state-of-the-art results in equation discovery, showing an absolute improvement of 7.1% over the best contender, by applying this ensemble method to 113 datasets within SRBench with known ground-truth equations.

Multi-Objective Bayesian Optimization with Active Preference Learning

There are a lot of real-world black-box optimization problems that need to optimize multiple criteria simultaneously. However, in a multi-objective optimization (MOO) problem, identifying the whole Pareto front requires the prohibitive search cost, while in many practical scenarios, the decision maker (DM) only needs a specific solution among the set of the Pareto optimal solutions. We propose a Bayesian optimization (BO) approach to identifying the most preferred solution in the MOO with expensive objective functions, in which a Bayesian preference model of the DM is adaptively estimated by an interactive manner based on the two types of supervisions called the pairwise preference and improvement request. To explore the most preferred solution, we define an acquisition function in which the uncertainty both in the objective function and the DM preference is incorporated. Further, to minimize the interaction cost with the DM, we also propose an active learning strategy for the preference estimation. We empirically demonstrate the effectiveness of our proposed method through the benchmark function optimization and the hyper-parameter optimization problems for machine learning models.

Generating Cheap Representative Functions for Expensive Automotive Crashworthiness Optimization

Solving real-world engineering optimization problems, such as automotive crashworthiness optimization, is extremely challenging, because the problem characteristics are oftentimes not well understood. Furthermore, typical hyperparameter optimization approaches that require a large function evaluation budget are computationally hindered, if the function evaluation is expensive, e.g., requires finite element (FE) simulation runs. In this paper, we propose an approach to characterize real-world expensive black-box optimization problems using the exploratory landscape analysis (ELA). Based on these landscape characteristics, we can identify test functions that are fast-to-evaluate and representative for hyperparameter optimization purposes. Focusing on 20 problem instances from automotive crashworthiness optimization, our results reveal that these 20 crashworthiness problems exhibit landscape features different from classical optimization benchmark test suites, such as the widely-used black-box optimization benchmarking (BBOB) problem set. In fact, these 20 problem instances belong to problem classes that are distinct from the BBOB test functions based on the clustering results. Further analysis indicates that, as far as the ELA features concern, they are most similar to problem classes of tree-based test functions. By analyzing the performance of two optimization algorithms with different hyperparameters, namely the covariance matrix adaptation evolutionary strategy (CMA-ES) and Bayesian optimization (BO), we show that the tree-based test functions are indeed representative in terms of predicting the algorithm performances. Following this, such scalable and fast-to-evaluate tree-based test functions have promising potential for automated design of an optimization algorithm for specific real-world problem classes.

Marginal Probability-Based Integer Handling for CMA-ES Tackling Single-and Multi-Objective Mixed-Integer Black-Box Optimization

This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The CMA-ES, our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous BBO. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing integer handling for the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple integer handling for the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method. Furthermore, in order to demonstrate the generality of the idea of the proposed method, in addition to the single-objective optimization case, we incorporate it into multi-objective CMA-ES and verify its performance on bi-objective mixed-integer benchmark problems.

Data-driven adaptive building thermal controller tuning with constraints: A primal–dual contextual Bayesian optimization approach

We study the problem of tuning the parameters of a room temperature controller to minimize its energy consumption, subject to the constraint that the daily cumulative thermal discomfort of the occupants is below a given threshold. We formulate it as an online constrained black-box optimization problem where, on each day, we observe some relevant environmental context and adaptively select the controller parameters. In this paper, we propose to use a data-driven Primal-Dual Contextual Bayesian Optimization (PDCBO) approach to solve this problem.In a simulation case study on a single room, we apply our algorithm to tune the parameters of a Proportional Integral (PI) heating controller and the pre-heating time. Our results show that PDCBO can save up to 4.7% energy consumption compared to other state-of-the-art Bayesian optimization-based methods while keeping the daily thermal discomfort below the given tolerable threshold on average. Additionally, PDCBO can automatically track time-varying tolerable thresholds while existing methods fail to do so. We then study an alternative constrained tuning problem where we aim to minimize the thermal discomfort with a given energy budget. With this formulation, PDCBO reduces the average discomfort by up to 63% compared to state-of-the-art safe optimization methods while keeping the average daily energy consumption below the required threshold.

Engine Calibration With Surrogate-Assisted Bilevel Evolutionary Algorithm.

Engine calibration problems are black-box optimization problems which are evaluation costly and most of them are constrained in the objective space. In these problems, decision variables may have different impacts on objectives and constraints, which could be detected by sensitivity analysis. Most existing surrogate-assisted evolutionary algorithms do not analyze variable sensitivity, thus, useless effort may be made on some less sensitive variables. This article proposes a surrogate-assisted bilevel evolutionary algorithm to solve a real-world engine calibration problem. Principal component analysis is performed to investigate the impact of variables on constraints and to divide decision variables into lower-level and upper-level variables. The lower-level aims at optimizing lower-level variables to make candidate solutions feasible, and the upper-level focuses on adjusting upper-level variables to optimize the objective. In addition, an ordinal-regression-based surrogate is adapted to estimate the ordinal landscape of solution feasibility. Computational studies on a gasoline engine model demonstrate that our algorithm is efficient in constraint handling and also achieves a smaller fuel consumption value than other state-of-the-art calibration methods.

Federated Many-Task Bayesian Optimization

Bayesian optimization is a powerful surrogate-assisted algorithm for solving expensive black-box optimization problems. While Bayesian optimization was developed for centralized optimization, the availability of massive distributed data has attracted increased interests in exploring federated Bayesian optimization that can use data on multiple clients without leaking the raw data. However, existing federated Bayesian optimization (FBO) approaches assume that either all clients jointly solve the same optimization task, or only one client solves one target optimization task by transferring knowledge from others in a federated way, making them unsuited for many real-world applications. In this paper, we consider FBO for the scenario where multiple related local black-box tasks associated with different clients are jointly optimized by sharing knowledge between tasks without leaking the data privacy. An efficient federated many-task Bayesian optimization framework is proposed to address not independent and identically distributed (non-IID) data while protecting the data privacy in the federated setting. A novel federated knowledge transfer paradigm is developed for dynamic many-task model aggregation according to a dissimilarity matrix. The dissimilarity is measured based on the rank of the predictions and only the hyperparameters in the local Gaussian process models are shared. In addition, a federated ensemble acquisition function is constructed by integrating the predictions of two surrogates using the global and local hyperparameters, respectively, to effectively search for the optimal solution. Experimental results show that our proposed method has reliable performance on both benchmark problems and a real machine learning problem also in the presence of non-IID data.

The comparison of different PDP-type self-adaptive schemes for the cooperation of GA, DE, and PSO algorithms

Many global optimization problems are presented as a black-box model, in which there is no information on the objective function properties. Traditional optimization algorithms usually can't effectively solve that kind of problems. Different heuristics and metaheuristics are usually applied in that case. Evolutionary algorithms are one of the most popular and effective approaches to black-box optimization problems. However, it's hard to choose one specific method that will solve the given problem better than other algorithms. For dealing with this issue, self-adaptive schemes are usually implemented. In this paper we have investigated the performance of different PDP-type adaptive schemes using such popular evolutionary-based algorithms as Genetic Algorithm, Differential Evolution, and Particle Swarm Optimization. The experimental results on a set of benchmark problems have shown that investigated schemes can improve the performance compared with the performance of a stand-alone evolutionary algorithm. At the same time the choice of a scheme and its parameters affect the results.

Speeding-Up Evolutionary Algorithms to Solve Black-Box Optimization Problems

Speeding-Up Evolutionary Algorithms to Solve Black-Box Optimization Problems

A solution method for mixed-variable constrained blackbox optimization problems

A solution method for mixed-variable constrained blackbox optimization problems

Optimizing additive manufacturing path pattern for Ti-6Al-4V thin rods using a combinatorial radial basis function surrogate-assisted genetic algorithm

Path pattern is one of the most significant parameters in the additive manufacturing (AM) process because it influences the specimen's final shape and residual stress distribution. Generally, the optimal path pattern is a computationally expensive, high-dimensional, and black-box permutation optimization problem. In this paper, we propose a combinatorial radial basis function surrogate-assisted genetic algorithm (CRBF-GA) to effectively generate the optimal path pattern by integrating the combinatorial radial basis function surrogate model (CRBF) with the genetic algorithm (GA). To demonstrate the effectiveness of the proposed CRBF-GA, a Ti-6Al-4V thin rod, a component of lattice, is chosen as the research object. Through numerical simulation, experimental verification, and error comparison analysis, the RBF-GA pattern is demonstrated to be the best path pattern among the random forest-assisted evolutionary algorithm (RF-EA), GA, spiral, and zigzag patterns, and it excels in achieving a more precise rod shape compared to the other patterns examined.

A Derivative-free Trust-region Method for Optimization on the Ellipsoid

Optimization methods play a crucial role in various fields and applications. In some optimization problems, the derivative information of the objective function is unavailable. Such black-box optimization problems need to be solved by derivative-free optimization methods. At the same time, optimization problems with ellipsoidal constraints are important and have widespread applications in various fields as well. Following the development of the late professor M. J. D. Powell’s efficient derivative-free trust-region optimization methods, this paper considers solving derivative-free optimization problems on the ellipsoid. Our new optimization solver EC-NEWUOA for problems on the ellipsoid in ℜ n is designed based on Powell’s derivative-free software NEWUOA for unconstrained optimization problems. The proposed techniques for our new method mainly include using the Courant penalty function, the augmented Lagrangian method, and the projection technique. Details about the method and theoretical analysis are included in this paper. We also compare our new method with other algorithms by solving test problems and then show the numerical advantages of our new method.

Automatic tuning of robust model predictive control in iterative tasks using efficient Bayesian optimization

Robust model predictive control (RMPC) is an effective technology for controlling uncertain systems while robustly handling constraints, and its closed-loop performance heavily relies on the selection of objective functions. However, the objective functions are typically chosen to be close to the real control objectives, despite an objective function that leads to less conservative constraints often provides better closed-loop performance. In this paper, we propose an automatic tuning framework for RMPC in iterative tasks. In particular, we parameterize RMPC and develop a Bayesian optimization (BO) method to tune it by solving a black-box optimization problem. We then introduce an efficient transfer learning framework within BO, which speeds up the searching process and enhances the controller performance. The effectiveness of the proposed tuning framework is illustrated on numerical examples.

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.