A Data Stream Ensemble Assisted Multifactorial Evolutionary Algorithm for Offline Data-Driven Dynamic Optimization

Existing work on offline data-driven optimization mainly focuses on problems in static environments, and little attention has been paid to problems in dynamic environments. Offline data-driven optimization in dynamic environments is a challenging problem because the distribution of collected data varies over time, requiring surrogate models and optimal solutions tracking with time. This paper proposes a knowledge-transfer-based data-driven optimization algorithm to address these issues. First, an ensemble learning method is adopted to train surrogate models to leverage the knowledge of data in historical environments as well as adapt to new environments. Specifically, given data in a new environment, a model is constructed with the new data, and the preserved models of historical environments are further trained with the new data. Then, these models are considered to be base learners and combined as an ensemble surrogate model. After that, all base learners and the ensemble surrogate model are simultaneously optimized in a multitask environment for finding optimal solutions for real fitness functions. In this way, the optimization tasks in the previous environments can be used to accelerate the tracking of the optimum in the current environment. Since the ensemble model is the most accurate surrogate, we assign more individuals to the ensemble surrogate than its base learners. Empirical results on six dynamic optimization benchmark problems demonstrate the effectiveness of the proposed algorithm compared with four state-of-the-art offline data-driven optimization algorithms. Code is available at https://github.com/Peacefulyang/DSE_MFS.git.

Evolutionary Algorithms for Parameter Optimization-Thirty Years Later

Thirty years, 1993-2023, is a huge time frame in science. We address some major developments in the field of evolutionary algorithms, with applications in parameter optimization, over these 30 years. These include the covariance matrix adaptation evolution strategy and some fast-growing fields such as multimodal optimization, surrogate-assisted optimization, multiobjective optimization, and automated algorithm design. Moreover, we also discuss particle swarm optimization and differential evolution, which did not exist 30 years ago, either. One of the key arguments made in the paper is that we need fewer algorithms, not more, which, however, is the current trend through continuously claiming paradigms from nature that are suggested to be useful as new optimization algorithms. Moreover, we argue that we need proper benchmarking procedures to sort out whether a newly proposed algorithm is useful or not. We also briefly discuss automated algorithm design approaches, including configurable algorithm design frameworks, as the proposed next step toward designing optimization algorithms automatically, rather than by hand.

On Nonstationary Gaussian Process Model for Solving Data-Driven Optimization Problems

In data-driven evolutionary optimization, most existing Gaussian processes (GPs)-assisted evolutionary algorithms (EAs) adopt stationary GPs (SGPs) as surrogate models, which might be insufficient for solving most optimization problems. This article finds that GPs in the optimization problems are nonstationary with great probability. We propose to employ a nonstationary GP (NSGP) surrogate model for data-driven evolutionary optimization, where the mean of the NSGP is allowed to vary with the decision variables, while its residue variance follows an SGP. In this article, the nonstationarity of GPs in the tested functions is theoretically analyzed. In addition, this article constructs an NSGP where the SGP is a degenerate case. Performance comparisons of the NSGP with the SGP and the NSGP-assisted EA (NSGP-MAEA) with the SGP-assisted EA (SGP-MAEA) are carried out on a set of benchmark problems and an antenna design problem. These comparison results demonstrate the competitiveness of the NSGP model.

Two-Stage Data-Driven Evolutionary Optimization for High-Dimensional Expensive Problems

Surrogate-assisted evolutionary algorithms (SAEAs) have been widely used for solving complex and computationally expensive optimization problems. However, most of the existing algorithms converge slowly in the later stage. This article proposes a novel two-stage data-driven evolutionary optimization (TS-DDEO) that meets the requirements of early exploration and later exploitation. In the first stage, a surrogate-assisted hierarchical particle swarm optimization method is used to find a promising area from the entire search space. In the second stage, we propose a best-data-driven optimization (BDDO) method with a strong exploitation ability to accelerate the optimization process. BDDO has a real-time update mechanism for the surrogate model and population and uses a predefined number of ranking-top solutions to update population and surrogates. BDDO combines three surrogate-assisted evolutionary sampling strategies: 1) surrogate-assisted differential evolution sampling; 2) surrogate-assisted local search; and 3) a surrogate-assisted full-crossover (FC) strategy which is proposed to integrate existing best genotypes in the population. Experiments and analysis have validated the effectiveness of the two-stage framework, the BDDO method, and the FC strategy. Moreover, the proposed algorithm is compared with five state-of-the-art SAEAs on high-dimensional benchmark functions. The result shows that TS-DDEO performs better both in effectiveness and robustness.

Bi-indicator driven surrogate-assisted multi-objective evolutionary algorithms for computationally expensive problems

This paper presents a bi-indicator-based surrogate-assisted evolutionary algorithm (BISAEA) for multi-objective optimization problems (MOPs) with computationally expensive objectives. In BISAEA, a Pareto-based bi-indictor strategy is proposed based on convergence and diversity indicators, where a nondominated sorting approach is adopted to carry out two-objective optimization (convergence and diversity indicators) problems. The radius-based function (RBF) models are used to approximate the objective values. In addition, the proposed algorithm adopts a one-by-one selection strategy to obtain promising samples from new samples for evaluating the true objectives by their angles and Pareto dominance relationship with real non-dominated solutions to improve the diversity. After the comparison with four state-of-the-art surrogate-assisted evolutionary algorithms and three evolutionary algorithms on 76 widely used benchmark problems, BISAEA shows high efficiency and a good balance between convergence and diversity. Finally, BISAEA is applied to the multidisciplinary optimization of blend-wing-body underwater gliders with 30 decision variables and three objectives, and the results demonstrate that BISAEA has superior performance on computationally expensive engineering problems.

Precision agriculture management based on a surrogate model assisted multiobjective algorithmic framework

Sustainable intensification needs to optimize irrigation and fertilization strategies while increasing crop yield. To enable more precision and effective agricultural management, a bi-level screening and bi-level optimization framework is proposed. Irrigation and fertilization dates are obtained by upper-level screening and upper-level optimization. Subsequently, due to the complexity of the problem, the lower-level optimization uses a data-driven evolutionary algorithm, which combines the fast non-dominated sorting genetic algorithm (NSGA-II), surrogate-assisted model of radial basis function and Decision Support System for Agrotechnology Transfer to handle the expensive objective problem and produce a set of optimal solutions representing a trade-off between conflicting objectives. Then, the lower-level screening quickly finds better irrigation and fertilization strategies among thousands of solutions. Finally, the experiment produces a better irrigation and fertilization strategy, with water consumption reduced by 44%, nitrogen application reduced by 37%, and economic benefits increased by 7 to 8%.

Variable surrogate model-based particle swarm optimization for high-dimensional expensive problems

Many industrial applications require time-consuming and resource-intensive evaluations of suitable solutions within very limited time frames. Therefore, many surrogate-assisted evaluation algorithms (SAEAs) have been widely used to optimize expensive problems. However, due to the curse of dimensionality and its implications, scaling SAEAs to high-dimensional expensive problems is still challenging. This paper proposes a variable surrogate model-based particle swarm optimization (called VSMPSO) to meet this challenge and extends it to solve 200-dimensional problems. Specifically, a single surrogate model constructed by simple random sampling is taken to explore different promising areas in different iterations. Moreover, a variable model management strategy is used to better utilize the current global model and accelerate the convergence rate of the optimizer. In addition, the strategy can be applied to any SAEA irrespective of the surrogate model used. To control the trade-off between optimization results and optimization time consumption of SAEAs, we consider fitness value and running time as a bi-objective problem. Applying the proposed approach to a benchmark test suite of dimensions ranging from 30 to 200 and comparisons with four state-of-the-art algorithms show that the proposed VSMPSO achieves high-quality solutions and computational efficiency for high-dimensional problems.

Evolutionary Computation for Expensive Optimization: A Survey

Expensive optimization problem (EOP) widely exists in various significant real-world applications. However, EOP requires expensive or even unaffordable costs for evaluating candidate solutions, which is expensive for the algorithm to find a satisfactory solution. Moreover, due to the fast-growing application demands in the economy and society, such as the emergence of the smart cities, the internet of things, and the big data era, solving EOP more efficiently has become increasingly essential in various fields, which poses great challenges on the problem-solving ability of optimization approach for EOP. Among various optimization approaches, evolutionary computation (EC) is a promising global optimization tool widely used for solving EOP efficiently in the past decades. Given the fruitful advancements of EC for EOP, it is essential to review these advancements in order to synthesize and give previous research experiences and references to aid the development of relevant research fields and real-world applications. Motivated by this, this paper aims to provide a comprehensive survey to show why and how EC can solve EOP efficiently. For this aim, this paper firstly analyzes the total optimization cost of EC in solving EOP. Then, based on the analysis, three promising research directions are pointed out for solving EOP, which are problem approximation and substitution, algorithm design and enhancement, and parallel and distributed computation. Note that, to the best of our knowledge, this paper is the first that outlines the possible directions for efficiently solving EOP by analyzing the total expensive cost. Based on this, existing works are reviewed comprehensively via a taxonomy with four parts, including the above three research directions and the real-world application part. Moreover, some future research directions are also discussed in this paper. It is believed that such a survey can attract attention, encourage discussions, and stimulate new EC research ideas for solving EOP and related real-world applications more efficiently.

A survey on evolutionary computation for complex continuous optimization

Complex continuous optimization problems widely exist nowadays due to the fast development of the economy and society. Moreover, the technologies like Internet of things, cloud computing, and big data also make optimization problems with more challenges including Many-dimensions, Many-changes, Many-optima, Many-constraints, and Many-costs. We term these as 5-M challenges that exist in large-scale optimization problems, dynamic optimization problems, multi-modal optimization problems, multi-objective optimization problems, many-objective optimization problems, constrained optimization problems, and expensive optimization problems in practical applications. The evolutionary computation (EC) algorithms are a kind of promising global optimization tools that have not only been widely applied for solving traditional optimization problems, but also have emerged booming research for solving the above-mentioned complex continuous optimization problems in recent years. In order to show how EC algorithms are promising and efficient in dealing with the 5-M complex challenges, this paper presents a comprehensive survey by proposing a novel taxonomy according to the function of the approaches, including reducing problem difficulty, increasing algorithm diversity, accelerating convergence speed, reducing running time, and extending application field. Moreover, some future research directions on using EC algorithms to solve complex continuous optimization problems are proposed and discussed. We believe that such a survey can draw attention, raise discussions, and inspire new ideas of EC research into complex continuous optimization problems and real-world applications.

Evolutionary Multiobjective Optimization Driven by Generative Adversarial Networks (GANs)

Recently, increasing works have been proposed to drive evolutionary algorithms using machine-learning models. Usually, the performance of such model-based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e., the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales due to the curse of dimensionality. To address this issue, we propose a multiobjective evolutionary algorithm driven by the generative adversarial networks (GANs). At each generation of the proposed algorithm, the parent solutions are first classified into real and fake samples to train the GANs; then the offspring solutions are sampled by the trained GANs. Thanks to the powerful generative ability of the GANs, our proposed algorithm is capable of generating promising offspring solutions in high-dimensional decision space with limited training data. The proposed algorithm is tested on ten benchmark problems with up to 200 decision variables. The experimental results on these test problems demonstrate the effectiveness of the proposed algorithm.

Evolutionary Optimisation for Reduction of the Low-Frequency Discrete-Spectrum Force of Marine Propeller Based on a Data-Driven Surrogate Model

For practical problems with non-convex, large-scale and highly constrained characteristics, evolutionary optimisation algorithms are widely used. However, advanced data-driven methods have yet to be comprehensively applied in related fields. In this study, a surrogate model combined with the Non-dominated Sorting Genetic Algorithm II-Differential Evolution (NSGA-II-DE) is applied to reduce the low-frequency Discrete-Spectrum (DS) force of propeller noise. Reduction of this force has drawn a lot of attention as it is the primary signal used in the sonar-based detection and identification of ships. In the present study, a surrogate model is proposed based on a trained Back-Propagation (BP) fully connected neural network, which improves the optimisation efficiency. The neural network is designed by analysing the depth and width of the hidden layers. The results indicate that a four-layer neural network with 64, 128, 256 and 64 nodes in each layer, respectively, exhibits the highest prediction accuracy. The prediction errors for the first order of DST, second order of DST and the thrust coefficient are only 0.21%, 5.71% and 0.01%, respectively. Data-Driven Evolutionary Optimisation (DDEO) is applied to a standard high-skew propeller to reduce DST. DDEO and a Traditional Evolutionary Optimisation Method (TEOM) obtain the same optimisation results, while the time cost of DDEO is only 0.68% that of the TEOM. Thus, the proposed DDEO is applicable to complex engineering problems in various fields.

A Random Forest-Assisted Evolutionary Algorithm for Data-Driven Constrained Multiobjective Combinatorial Optimization of Trauma Systems

Many real-world optimization problems can be solved by using the data-driven approach only, simply because no analytic objective functions are available for evaluating candidate solutions. In this paper, we address a class of expensive data-driven constrained multiobjective combinatorial optimization problems, where the objectives and constraints can be calculated only on the basis of a large amount of data. To solve this class of problems, we propose using random forests (RFs) and radial basis function networks as surrogates to approximate both objective and constraint functions. In addition, logistic regression models are introduced to rectify the surrogate-assisted fitness evaluations and a stochastic ranking selection is adopted to further reduce the influences of the approximated constraint functions. Three variants of the proposed algorithm are empirically evaluated on multiobjective knapsack benchmark problems and two real-world trauma system design problems. Experimental results demonstrate that the variant using RF models as the surrogates is effective and efficient in solving data-driven constrained multiobjective combinatorial optimization problems.

Heterogeneous Ensemble-Based Infill Criterion for Evolutionary Multiobjective Optimization of Expensive Problems

Gaussian processes (GPs) are the most popular model used in surrogate-assisted evolutionary optimization of computationally expensive problems, mainly because GPs are able to measure the uncertainty of the estimated fitness values, based on which certain infill sampling criteria can be used to guide the search and update the surrogate model. However, the computation time for constructing GPs may become excessively long when the number of training samples increases, which makes it inappropriate to use them as surrogates in evolutionary optimization. To address this issue, this paper proposes to use ensembles as surrogates and infill criteria for model management in evolutionary optimization. A heterogeneous ensemble consisting of a least square support vector machine and two radial basis function networks is constructed to enhance the reliability of ensembles for uncertainty estimation. In addition to the original decision variables, a selected subset of the decision variables and a set of transformed variables are used as inputs of the heterogeneous ensemble to further promote the diversity of the ensemble. The proposed heterogeneous ensemble is compared with a GP and a homogeneous ensemble for infill sampling criteria in evolutionary multiobjective optimization. Experimental results demonstrate that the heterogeneous ensemble is competitive in performance compared with GPs and much more scalable in computational complexity to the increase in search dimension.

Population Diversity of Particle Swarm Optimizer Solving Single and Multi-Objective Problems

Premature convergence occurs in swarm intelligence algorithms searching for optima. A swarm intelligence algorithm has two kinds of abilities: exploration of new possibilities and exploitation of old certainties. The exploration ability means that an algorithm can explore more search place to increase the possibility that the algorithm can find good enough solutions. In contrast, the exploitation ability means that an algorithm focuses on the refinement of found promising areas. An algorithm should have a balance between exploration and exploitation, that is, the allocation of computational resources should be optimized to ensure that an algorithm can find good enough solutions effectively. The diversity measures the distribution of individuals’ information. From the observation of the distribution and diversity change, the degree of exploration and exploitation can be obtained. Another issue in multiobjective is the solution metric. Pareto domination is utilized to compare between two solutions, however, solutions are almost Pareto non-dominated for multi-objective problems with more than ten objectives. In this paper, the authors analyze the population diversity of particle swarm optimizer for solving both single objective and multiobjective problems. The population diversity of solutions is used to measure the goodness of a set of solutions. This metric may guide the search in problems with numerous objectives. Adaptive optimization algorithms can be designed through controlling the balance between exploration and exploitation.