Large-scale Black-box Optimization 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.

SF-FWA: A Self-Adaptive Fast Fireworks Algorithm for effective large-scale optimization

Computationally efficient algorithms for large-scale black-box optimization have become increasingly important in recent years due to the growing complexity of engineering and scientific problems. In this paper, a novel algorithm called the Self-adaptive Fast Fireworks Algorithm (SF-FWA) is proposed to effectively conduct large-scale black-box optimization. The main idea is to utilize a set of expressive and computationally efficient search distributions to cope with different function landscapes while tuning the hyperparameters of the search distributions in an online fashion. To achieve this, the Expressive Fast Explosion (EFE) mechanism is designed to achieve effective and efficient sampling, and the Inter-Fireworks Competitive Cooperation (IFCC) mechanism is designed to adapt hyperparameter distributions. This new optimization paradigm equips the population with the ability to automatically adjust to a rich set of function landscapes with linear computational complexity in terms of problem dimensionality. Experimental studies show that SF-FWA can not only exploit the separability of the problem efficiently but can also deal with rotational transformations to the coordinate system. The numerical results on the standard large-scale optimization benchmark suite indicate that SF-FWA outperforms current state-of-the-art large-scale optimization algorithms. The outstanding performance of SF-FWA on optimizing neural network controllers for solving reinforcement learning tasks demonstrates its great potential to be applied to a wider range of real-world problems.

An Efficient Adaptive Differential Grouping Algorithm for Large-Scale Black-Box Optimization

Decomposition plays a significant role in cooperative coevolution, which shows great potential in large-scale black-box optimization (LSBO). However, current learning-based decomposition algorithms require many fitness evaluations (FEs) to detect variable interdependencies and encounter the difficulty of threshold setting. To address these issues, this study proposes an efficient adaptive differential grouping (EADG) algorithm. Instead of homogeneously tackling different types of LSBO instances, EADG first identifies the instance type by detecting the interdependencies of a few pairs of variable subsets. Only if the instance is partially separable dose EADG further engage with it by converting its decomposition process into a search process in a binary tree. This facilitates the systematic reutilization of evaluated solutions so that half the interdependencies can be directly deduced without extra FEs. To promote the decomposition accuracy, EADG specially designs a normalized interdependency indicator that can adaptively generate a decomposition threshold according to its ordinal distribution. Theoretical analysis and experimental results show that EADG outperforms current popular decomposition algorithms. Further tests indicate that it can help cooperative coevolution achieve highly competitive optimization performance.

Advances in meta-heuristic methods for large-scale black-box optimization problems

The optimal design of complex engineering equipment usually faces high-complexity, high-dimensional optimization problems – the so-called black-box optimization problems (LBOPs) – which are characterized by unavailable mathematical expressions of objective functions and/or constraint functions, and high dimensionality of design variables. The LBOPs have attracted the interest of scholars in various fields in recent years, and meta-heuristic algorithms are considered effective methods for solving these problems. This paper comprehensively summarizes recent research progress in meta-heuristic algorithms for solving LBOPs, including meta-heuristic algorithms with and without decomposition strategies, and meta-heuristic algorithms for handling computationally expensive large-scale optimization problems. Finally, possible future research directions of meta-heuristic methods for solving LBOPs are proposed.

A topology-based single-pool decomposition framework for large-scale global optimization

Identification of variable interaction plays a crucial role in applying a divide-and-conquer algorithm for large-scale black-box optimization. However, most of the existing decomposition methods are less efficient in decomposing the overlapping problems. This drawback diminishes the practicality of the existing methods. In this paper, we propose an efficient single-pool decomposition framework (SPDF). The interactions of decision variables are identified in an ordinal fashion. The unbalanced grouping efficiency of the existing decomposition methods can be significantly alleviated. Furthermore, we find that the grouping efficiency can be further improved by integrating the topological information into the decomposition process. In many real-world problems, this information can be 1-, 2- or 3-dimensional coordinates, which represent the geometric structure of the large-scale systems. Based on this, we propose a topology-based decomposition method, which we call Topology-based Single-Pool Differential Grouping (TSPDG). The efficacy of our proposed methods is demonstrated on the CEC’2010 and the CEC’2013 large-scale benchmark suites, as well as a practical case study in production optimization.

Large Scale Black-Box Optimization by Limited-Memory Matrix Adaptation

The covariance matrix adaptation evolution strategy (CMA-ES) is a popular method to deal with nonconvex and/or stochastic optimization problems when gradient information is not available. Being based on the CMA-ES, the recently proposed matrix adaptation evolution strategy (MA-ES) establishes the rather surprising result that the covariance matrix and all associated operations (e.g., potentially unstable eigen decomposition) can be replaced by an iteratively updated transformation matrix without any loss of performance. In order to further simplify MA-ES and reduce its $\boldsymbol {\mathcal {O}}({n}^{ { {2}}})$ time and storage complexity to $\boldsymbol {\mathcal {O}}({{mn}})$ with $\boldsymbol {m}~{\boldsymbol \ll }~{n}$ such as $\boldsymbol {m}~{\boldsymbol \in }~\boldsymbol {\mathcal {O}}{(1)}$ or ${m} {\in } \boldsymbol {\mathcal {O}}({\log {({n})}})$ , we present the limited-memory MA-ES for efficient zeroth order large-scale optimization. The algorithm demonstrates state-of-the-art performance on a set of established large-scale benchmarks.

Fast Covariance Matrix Adaptation for Large-Scale Black-Box Optimization.

Covariance matrix adaptation evolution strategy (CMA-ES) is a successful gradient-free optimization algorithm. Yet, it can hardly scale to handle high-dimensional problems. In this paper, we propose a fast variant of CMA-ES (Fast CMA-ES) to handle large-scale black-box optimization problems. We approximate the covariance matrix by a low-rank matrix with a few vectors and use two of them to generate each new solution. The algorithm achieves linear internal complexity on the dimension of search space. We illustrate that the covariance matrix of the underlying distribution can be considered as an ensemble of simple models constructed by two vectors. We experimentally investigate the algorithm's behaviors and performances. It is more efficient than the CMA-ES in terms of running time. It outperforms or performs comparatively to the variant limited memory CMA-ES on large-scale problems. Finally, we evaluate the algorithm's performance with a restart strategy on the CEC'2010 large-scale global optimization benchmarks, and it shows remarkable performance and outperforms the large-scale variants of the CMA-ES.

Improved Differential Evolution for Large-Scale Black-Box Optimization

The demand for solving large-scale complex problems continues to grow. Many real-world problems are described by a large number of variables that interact with each other in a complex way. The dimensionality of the problem has a direct impact on the computational cost of the optimization. During the last two decades, differential evolution has been shown to be one of the most powerful optimizers for a wide range of optimization problems. In this paper, we investigate its appropriateness for large-scale problems. We propose a new variation of differential evolution that exhibits good results on difficult functions with a large numbers of variables. The proposed algorithm incorporates the following mechanisms: the use of three strategies, the extended range of values for self-adapted parameters $F$ and $\mathit {CR}$ , subpopulations, and the population size reduction. The algorithm was tested on the CEC 2013 benchmark suite for large-scale optimization, and on two real-world problems from the CEC 2011 benchmark suite on real-world optimization. A comparative analysis was performed with recently proposed algorithms. The analysis shows the superior performance of our algorithm on most complex problems, described by overlapping and non-separable functions.

DG2: A Faster and More Accurate Differential Grouping for Large-Scale Black-Box Optimization

Identification of variable interaction is essential for an efficient implementation of a divide-and-conquer algorithm for large-scale black-box optimization. In this paper, we propose an improved variant of the differential grouping (DG) algorithm, which has a better efficiency and grouping accuracy. The proposed algorithm, DG2, finds a reliable threshold value by estimating the magnitude of roundoff errors. With respect to efficiency, DG2 reuses the sample points that are generated for detecting interactions and saves up to half of the computational resources on fully separable functions. We mathematically show that the new sampling technique achieves the lower bound with respect to the number of function evaluations. Unlike its predecessor, DG2 checks all possible pairs of variables for interactions and has the capacity to identify overlapping components of an objective function. On the accuracy aspect, DG2 outperforms the state-of-the-art decomposition methods on the latest large-scale continuous optimization benchmark suites. DG2 also performs reliably in the presence of imbalance among contribution of components in an objective function. Another major advantage of DG2 is the automatic calculation of its threshold parameter ( $\epsilon $ ), which makes it parameter-free. Finally, the experimental results show that when DG2 is used within a cooperative co-evolutionary framework, it can generate competitive results as compared to several state-of-the-art algorithms.

Embedded Bandits for Large-Scale Black-Box Optimization

Random embedding has been applied with empirical success to large-scale black-box optimization problems with low effective dimensions. This paper proposes the EmbeddedHunter algorithm, which incorporates the technique in a hierarchical stochastic bandit setting, following the optimism in the face of uncertainty principle and breaking away from the multiple-run framework in which random embedding has been conventionally applied similar to stochastic black-box optimization solvers. Our proposition is motivated by the bounded mean variation in the objective value for a low-dimensional point projected randomly into the decision space of Lipschitz-continuous problems. In essence, the EmbeddedHunter algorithm expands optimistically a partitioning tree over a low-dimensional — equal to the effective dimension of the problem —search space based on a bounded number of random embeddings of sampled points from the low-dimensional space. In contrast to the probabilistic theoretical guarantees of multiple-run random-embedding algorithms, the finite-time analysis of the proposed algorithm presents a theoretical upper bound on the regret as a function of the algorithm's number of iterations. Furthermore, numerical experiments were conducted to validate its performance. The results show a clear performance gain over recently proposed random embedding methods for large-scale problems, provided the intrinsic dimensionality is low.

A Competitive Divide-and-Conquer Algorithm for Unconstrained Large-Scale Black-Box Optimization

This article proposes a competitive divide-and-conquer algorithm for solving large-scale black-box optimization problems for which there are thousands of decision variables and the algebraic models of the problems are unavailable. We focus on problems that are partially additively separable, since this type of problem can be further decomposed into a number of smaller independent subproblems. The proposed algorithm addresses two important issues in solving large-scale black-box optimization: (1) the identification of the independent subproblems without explicitly knowing the formula of the objective function and (2) the optimization of the identified black-box subproblems. First, a Global Differential Grouping (GDG) method is proposed to identify the independent subproblems. Then, a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is adopted to solve the subproblems resulting from its rotation invariance property. GDG and CMA-ES work together under the cooperative co-evolution framework. The resultant algorithm, named CC-GDG-CMAES, is then evaluated on the CEC’2010 large-scale global optimization (LSGO) benchmark functions, which have a thousand decision variables and black-box objective functions. The experimental results show that, on most test functions evaluated in this study, GDG manages to obtain an ideal partition of the index set of the decision variables, and CC-GDG-CMAES outperforms the state-of-the-art results. Moreover, the competitive performance of the well-known CMA-ES is extended from low-dimensional to high-dimensional black-box problems.

LM-CMA: An Alternative to L-BFGS for Large-Scale Black Box Optimization.

Limited-memory BFGS (L-BFGS; Liu and Nocedal, 1989 ) is often considered to be the method of choice for continuous optimization when first- or second-order information is available. However, the use of L-BFGS can be complicated in a black box scenario where gradient information is not available and therefore should be numerically estimated. The accuracy of this estimation, obtained by finite difference methods, is often problem-dependent and may lead to premature convergence of the algorithm. This article demonstrates an alternative to L-BFGS, the limited memory covariance matrix adaptation evolution strategy (LM-CMA) proposed by Loshchilov ( 2014 ). LM-CMA is a stochastic derivative-free algorithm for numerical optimization of nonlinear, nonconvex optimization problems. Inspired by L-BFGS, LM-CMA samples candidate solutions according to a covariance matrix reproduced from m direction vectors selected during the optimization process. The decomposition of the covariance matrix into Cholesky factors allows reducing the memory complexity to [Formula: see text], where n is the number of decision variables. The time complexity of sampling one candidate solution is also [Formula: see text] but scales as only about 25 scalar-vector multiplications in practice. The algorithm has an important property of invariance with respect to strictly increasing transformations of the objective function; such transformations do not compromise its ability to approach the optimum. LM-CMA outperforms the original CMA-ES and its large-scale versions on nonseparable ill-conditioned problems with a factor increasing with problem dimension. Invariance properties of the algorithm do not prevent it from demonstrating a comparable performance to L-BFGS on nontrivial large-scale smooth and nonsmooth optimization problems.