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

An Ensemble Surrogate-Based Coevolutionary Algorithm for Solving Large-Scale Expensive Optimization Problems

Surrogate-assisted evolutionary algorithms (SAEAs) have shown promising performance for solving expensive optimization problems (EOPs) whose true evaluations are computationally or physically expensive. However, most existing SAEAs only focus on the problems with low dimensionality and they rarely consider solving large-scale EOPs (LSEOPs). To fill this research gap, this article proposes an ensemble surrogate-based coevolutionary optimizer for tackling LSEOPs. First, some local surrogate models are trained with low-dimensional data subsets by using feature selection on the large-scale decision variables, a part of which are used to build a selective ensemble surrogate for better approximating the target LSEOP. Then, a coevolutionary optimizer guided by the ensemble surrogate is designed by running two populations to cooperatively solve the target LSEOP and the simplified auxiliary problem. The information of offspring from the two populations is shared to facilitate the coevolution process, which can exploit the searching experience from the simplified auxiliary problem to help solving the target LSEOP. Finally, an effective infill selection criterion is used to update the ensemble surrogate and enhance its approximate performance. To evaluate the performance of the proposed algorithm, a number of well-known benchmark problems are used and the experimental results validate our superior performance over nine state-of-the-art SAEAs on most cases.

A two-stage dominance-based surrogate-assisted evolution algorithm for high-dimensional expensive multi-objective optimization

In the past decades, surrogate-assisted evolutionary algorithms (SAEAs) have become one of the most popular methods to solve expensive multi-objective optimization problems (EMOPs). However, most existing methods focus on low-dimensional EMOPs because a large number of training samples are required to build accurate surrogate models, which is unrealistic for high-dimensional EMOPs. Therefore, this paper develops a two-stage dominance-based surrogate-assisted evolution algorithm (TSDEA) for high-dimensional EMOPs which utilizes the RBF model to approximate each objective function. First, a two-stage selection strategy is applied to select individuals for re-evaluation. Then considering the training time of the model, proposing a novel archive updating strategy to limit the number of individuals for updating. Experimental results show that the proposed algorithm has promising performance and computational efficiency compared to the state-of-the-art five SAEAs.

Multisurrogate-Assisted Multitasking Particle Swarm Optimization for Expensive Multimodal Problems

Many real-world applications can be formulated as expensive multimodal optimization problems (EMMOPs). When surrogate-assisted evolutionary algorithms (SAEAs) are employed to tackle these problems, they not only face the problem of selecting surrogate models but also need to tackle the problem of discovering and updating multiple modalities. Different optimization problems and different stages of evolutionary algorithms (EAs) generally require different types of surrogate models. To address this issue, in this article, we present a multisurrogate-assisted multitasking particle swarm optimization algorithm to seek multiple optimal solutions of EMMOPs at a low computational cost. The proposed algorithm first transforms an EMMOP into a multitasking optimization problem by integrating various surrogate models, and designs a multitasking niche particle swarm algorithm to solve it. Following that, a surrogate model management strategy based on the skill factor and clustering is developed to effectively balance the number of real function evaluations and the prediction accuracy of candidate optimal solutions. In addition, an adaptive local search strategy based on the trust region is proposed to enhance the capability of swarm in exploiting potential optimal modalities. We compare the proposed algorithm with five state-of-the-art SAEAs and seven multimodal EAs on 19 benchmark functions and the building energy conservation problem and experimental results show that the proposed algorithm can obtain multiple highly competitive optimal solutions.

A Surrogate-Assisted Differential Evolution Algorithm for High-Dimensional Expensive Optimization Problems

The radial basis function (RBF) model and the Kriging model have been widely used in the surrogate-assisted evolutionary algorithms (SAEAs). Based on their characteristics, a global and local surrogate-assisted differential evolution algorithm (GL-SADE) for high-dimensional expensive problems is proposed in this article, in which a global RBF model is trained with all samples to estimate a global trend, and then its optima is used to significantly accelerate the convergence process. A local Kriging model prefers to select points with good predicted fitness and great uncertainty, which can effectively prevent the search from getting trapped into local optima. When the local Kriging model finds the best solution so far, a reward search strategy is executed to further exploit the local Kriging model. The experiments on a set of benchmark functions with dimensions varying from 30 to 200 are conducted to evaluate the performance of the proposed algorithm. The experimental results of the proposed algorithm are compared to four state-of-the-art algorithms to show its effectiveness and efficiency in solving high-dimensional expensive problems. Besides, GL-SADE is applied to an airfoil optimization problem to show its effectiveness.

An Uncertainty Measure for Prediction of Non-Gaussian Process Surrogates

Model management is an essential component in data-driven surrogate-assisted evolutionary optimization. In model management, the solutions with a large degree of uncertainty in approximation play an important role. They can strengthen the exploration ability of algorithms and improve the accuracy of surrogates. However, there is no theoretical method to measure the uncertainty of prediction of Non-Gaussian process surrogates. To address this issue, this article proposes a method to measure the uncertainty. In this method, a stationary random field with a known zero mean is used to measure the uncertainty of prediction of Non-Gaussian process surrogates. Based on experimental analyses, this method is able to measure the uncertainty of prediction of Non-Gaussian process surrogates. The method's effectiveness is demonstrated on a set of benchmark problems in single surrogate and ensemble surrogates cases.

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.

Choose Appropriate Subproblems for Collaborative Modeling in Expensive Multiobjective Optimization

In dealing with the expensive multiobjective optimization problem, some algorithms convert it into a number of single-objective subproblems for optimization. At each iteration, these algorithms conduct surrogate-assisted optimization on one or multiple subproblems. However, these subproblems may be unnecessary or resolved. Operating on such subproblems can cause server inefficiencies, especially in the case of expensive optimization. To overcome this shortcoming, we propose an adaptive subproblem selection (ASS) strategy to identify the most promising subproblems for further modeling. To better leverage the cross information between the subproblems, we use the collaborative multioutput Gaussian process surrogate to model them jointly. Moreover, the commonly used acquisition functions (also known as infill criteria) are investigated in this article. Our analysis reveals that these acquisition functions may cause severe imbalances between exploitation and exploration in multiobjective optimization scenarios. Consequently, we develop a new acquisition function, namely, adaptive lower confidence bound (ALCB), to cope with it. The experimental results on three different sets of benchmark problems indicate that our proposed algorithm is competitive. Beyond that, we also quantitatively validate the effectiveness of the ASS strategy, the CoMOGP model, and the ALCB acquisition function.

Bridging the Micro and Macro: Calibration of Agent-Based Model Using Mean-Field Dynamics

Calibration of agent-based models (ABM) is an essential stage when they are applied to reproduce the actual behaviors of distributed systems. Unlike traditional methods that suffer from the repeated trial and error and slow convergence of iteration, this article proposes a new ABM calibration approach by establishing a link between agent microbehavioral parameters and systemic macro-observations. With the assumption that the agent behavior can be formulated as a high-order Markovian process, the new approach starts with a search for an optimal transfer probability through a macrostate transfer equation. Then, each agent's microparameter values are computed using mean-field approximation, where his complex dependencies with others are approximated by an expected aggregate state. To compress the agent state space, principal component analysis is also introduced to avoid high dimensions of the macrostate transfer equation. The proposed method is validated in two scenarios: 1) population evolution and 2) urban travel demand analysis. Experimental results demonstrate that compared with the machine-learning surrogate and evolutionary optimization, our method can achieve higher accuracies with much lower computational complexities.

Multisurrogate-Assisted Ant Colony Optimization for Expensive Optimization Problems With Continuous and Categorical Variables

As an effective optimization tool for expensive optimization problems (EOPs), surrogate-assisted evolutionary algorithms (SAEAs) have been widely studied in recent years. However, most current SAEAs are designed for continuous/ combinatorial EOPs, which are not suitable for mixed-variable EOPs. This article focuses on one kind of mixed-variable EOP: EOPs with continuous and categorical variables (EOPCCVs). A multisurrogate-assisted ant colony optimization algorithm (MiSACO) is proposed to solve EOPCCVs. MiSACO contains two main strategies: 1) multisurrogate-assisted selection and 2) surrogate-assisted local search. In the former, the radial basis function (RBF) and least-squares boosting tree (LSBT) are employed as the surrogate models. Afterward, three selection operators (i.e., RBF-based selection, LSBT-based selection, and random selection) are devised to select three solutions from the offspring solutions generated by ACO, with the aim of coping with different types of EOPCCVs robustly and preventing the algorithm from being misled by inaccurate surrogate models. In the latter, sequence quadratic optimization, coupled with RBF, is utilized to refine the continuous variables of the best solution found so far. By combining these two strategies, MiSACO can solve EOPCCVs with limited function evaluations. Three sets of test problems and two real-world cases are used to verify the effectiveness of MiSACO. The results demonstrate that MiSACO performs well in solving EOPCCVs.

When Does Diversity Help Generalization in Classification Ensembles?

Ensembles, as a widely used and effective technique in the machine learning community, succeed within a key element-"diversity." The relationship between diversity and generalization, unfortunately, is not entirely understood and remains an open research issue. To reveal the effect of diversity on the generalization of classification ensembles, we investigate three issues on diversity, that is, the measurement of diversity, the relationship between the proposed diversity and the generalization error, and the utilization of this relationship for ensemble pruning. In the diversity measurement, we measure diversity by error decomposition inspired by regression ensembles, which decompose the error of classification ensembles into accuracy and diversity. Then, we formulate the relationship between the measured diversity and ensemble performance through the theorem of margin and generalization and observe that the generalization error is reduced effectively only when the measured diversity is increased in a few specific ranges, while in other ranges, larger diversity is less beneficial to increasing the generalization of an ensemble. Besides, we propose two pruning methods based on diversity management to utilize this relationship, which could increase diversity appropriately and shrink the size of the ensemble without much-decreasing performance. The empirical results validate the reasonableness of the proposed relationship between diversity and ensemble generalization error and the effectiveness of the proposed pruning methods.

A Three-Level Radial Basis Function Method for Expensive Optimization

This article proposes a three-level radial basis function (TLRBF)-assisted optimization algorithm for expensive optimization. It consists of three search procedures at each iteration: 1) the global exploration search is to find a solution by optimizing a global RBF approximation function subject to a distance constraint in the whole search space; 2) the subregion search is to generate a solution by minimizing an RBF approximation function in a subregion determined by fuzzy clustering; and 3) the local exploitation search is to generate a solution by solving a local RBF approximation model in the neighborhood of the current best solution. Compared with some other state-of-the-art algorithms on five commonly used scalable benchmark problems, ten CEC2015 computationally expensive problems, and a real-world airfoil design optimization problem, our proposed algorithm performs well for expensive optimization.

Novel Multitask Conditional Neural-Network Surrogate Models for Expensive Optimization

Multiple-related tasks can be learned simultaneously by sharing information among tasks to avoid tabula rasa learning and to improve performance in the no transfer case (i.e., when each task learns in isolation). This study investigates multitask learning with conditional neural process (CNP) networks and proposes two multitask learning network models on the basis of CNPs, namely, the one-to-many multitask CNP (OMc-MTCNP) and the many-to-many MTCNP (MMc-MTCNP). Compared with existing multitask models, the proposed models add an extensible correlation learning layer to learn the correlation among tasks. Moreover, the proposed multitask CNP (MTCNP) networks are regarded as surrogate models and applied to a Bayesian optimization framework to replace the Gaussian process (GP) to avoid the complex covariance calculation. The proposed Bayesian optimization framework simultaneously infers multiple tasks by utilizing the possible dependencies among them to share knowledge across tasks. The proposed surrogate models augment the observed dataset with a number of related tasks to estimate model parameters confidently. The experimental studies under several scenarios indicate that the proposed algorithms are competitive in performance compared with GP-, single-task-, and other multitask model-based Bayesian optimization methods.

On Iterative Proportional Updating: Limitations and Improvements for General Population Synthesis

Population synthesis is the foundation of the agent-based social simulation. Current approaches mostly consider basic population and households, rather than other social organizations. This article starts with a theoretical analysis of the iterative proportional updating (IPU) algorithm, a representative method in this field, and then gives an extension to consider more social organization types. The IPU method, for the first time, proves to be unable to converge to an optimal population distribution that simultaneously satisfies the constraints from individual and household levels. It is further improved to a bilevel optimization, which can solve such a problem and include more than one type of social organization. Numerical simulations, as well as population synthesis using actual Chinese nationwide census data, support our theoretical conclusions and indicate that our proposed bilevel optimization can both synthesize more social organization types and get more accurate results.

TiDEC: A Two-Layered Integrated Decision Cycle for Population Evolution

Agent-based simulation is a useful approach for the analysis of dynamic population evolution. In this field, the existing models mostly treat the migration behavior as a result of utility maximization, which partially ignores the endogenous mechanisms of human decision making. To simulate such a process, this article proposes a new cognitive architecture called the two-layered integrated decision cycle (TiDEC) which characterizes the individual's decision-making process. Different from the previous ones, the new hybrid architecture incorporates deep neural networks for its perception and implicit knowledge learning. The proposed model is applied in China and U.S. population evolution. To the best of our knowledge, this is the first time that the cognitive computation is used in such a field. Computational experiments using the actual census data indicate that the cognitive model, compared with the traditional utility maximization methods, cannot only reconstruct the historical demographic features but also achieve better prediction of future evolutionary dynamics.

A multi-model assisted differential evolution algorithm for computationally expensive optimization problems

Surrogate models are commonly used to reduce the number of required expensive fitness evaluations in optimizing computationally expensive problems. Although many competitive surrogate-assisted evolutionary algorithms have been proposed, it remains a challenging issue to develop an effective model management strategy to address problems with different landscape features under a limited computational budget. This paper adopts a coarse-to-fine evaluation scheme basing on two surrogate models, i.e., a coarse Gaussian process and a fine radial basis function, for assisting a differential evolution algorithm to solve computationally expensive optimization problems. The coarse Gaussian process model is meant to capture the general contour of the fitness landscape to estimate the fitness and its degree of uncertainty. A surrogate-assisted environmental selection strategy is then developed according to the non-dominance relationship between approximated fitness and estimated uncertainty. Meanwhile, the fine radial basis function model aims to learn the details of the local fitness landscape to refine the approximation quality of the new parent population and find the local optima for real-evaluations. The performance and scalability of the proposed method are extensively evaluated on two sets of widely used benchmark problems. Experimental results show that the proposed method can outperform several state-of-the-art algorithms within a limited computational budget.

A federated data-driven evolutionary algorithm for expensive multi-/many-objective optimization

Data-driven optimization has found many successful applications in the real world and received increased attention in the field of evolutionary optimization. Most existing algorithms assume that the data used for optimization are always available on a central server for construction of surrogates. This assumption, however, may fail to hold when the data must be collected in a distributed way and are subject to privacy restrictions. This paper aims to propose a federated data-driven evolutionary multi-/many-objective optimization algorithm. To this end, we leverage federated learning for surrogate construction so that multiple clients collaboratively train a radial-basis-function-network as the global surrogate. Then a new federated acquisition function is proposed for the central server to approximate the objective values using the global surrogate and estimate the uncertainty level of the approximated objective values based on the local models. The performance of the proposed algorithm is verified on a series of multi-/many-objective benchmark problems by comparing it with two state-of-the-art surrogate-assisted multi-objective evolutionary algorithms.

Data-Driven Evolutionary Algorithm With Perturbation-Based Ensemble Surrogates

Data-driven evolutionary algorithms (DDEAs) aim to utilize data and surrogates to drive optimization, which is useful and efficient when the objective function of the optimization problem is expensive or difficult to access. However, the performance of DDEAs relies on their surrogate quality and often deteriorates if the amount of available data decreases. To solve these problems, this article proposes a new DDEA framework with perturbation-based ensemble surrogates (DDEA-PES), which contain two efficient mechanisms. The first is a diverse surrogate generation method that can generate diverse surrogates through performing data perturbations on the available data. The second is a selective ensemble method that selects some of the prebuilt surrogates to form a final ensemble surrogate model. By combining these two mechanisms, the proposed DDEA-PES framework has three advantages, including larger data quantity, better data utilization, and higher surrogate accuracy. To validate the effectiveness of the proposed framework, this article provides both theoretical and experimental analyses. For the experimental comparisons, a specific DDEA-PES algorithm is developed as an instance by adopting a genetic algorithm as the optimizer and radial basis function neural networks as the base models. The experimental results on widely used benchmarks and an aerodynamic airfoil design real-world optimization problem show that the proposed DDEA-PES algorithm outperforms some state-of-the-art DDEAs. Moreover, when compared with traditional nondata-driven methods, the proposed DDEA-PES algorithm only requires about 2% computational budgets to produce competitive results.

End-to-End Ensemble Learning by Exploiting the Correlation Between Individuals and Weights

Ensemble learning performs better than a single classifier in most tasks due to the diversity among multiple classifiers. However, the enhancement of the diversity is at the expense of reducing the accuracies of individual classifiers in general and, thus, how to balance the diversity and accuracies is crucial for improving the ensemble performance. In this paper, we propose a new ensemble method which exploits the correlation between individual classifiers and their corresponding weights by constructing a joint optimization model to achieve the tradeoff between the diversity and the accuracy. Specifically, the proposed framework can be modeled as a shallow network and efficiently trained by the end-to-end manner. In the proposed ensemble method, not only can a high total classification performance be achieved by the weighted classifiers but also the individual classifier can be updated based on the error of the optimized weighted classifiers ensemble. Furthermore, the sparsity constraint is imposed on the weight to enforce that partial individual classifiers are selected for final classification. Finally, the experimental results on the UCI datasets demonstrate that the proposed method effectively improves the performance of classification compared with relevant existing ensemble methods.

An Adaptive Heterogeneous Online Learning Ensemble Classifier for Nonstationary Environments

In recent years, the prevalence of technological advances has led to an enormous and ever-increasing amount of data that are now commonly available in a streaming fashion. In such nonstationary environments, the underlying process generating the data stream is characterized by an intrinsic nonstationary or evolving or drifting phenomenon known as concept drift. Given the increasingly common applications whose data generation mechanisms are susceptible to change, the need for effective and efficient algorithms for learning from and adapting to evolving or drifting environments can hardly be overstated. In dynamic environments associated with concept drift, learning models are frequently updated to adapt to changes in the underlying probability distribution of the data. A lot of work in the area of learning in nonstationary environments focuses on updating the learning predictive model to optimize recovery from concept drift and convergence to new concepts by adjusting parameters and discarding poorly performing models while little effort has been dedicated to investigate what type of learning model is suitable at any given time for different types of concept drift. In this paper, we investigate the impact of heterogeneous online ensemble learning based on online model selection for predictive modeling in dynamic environments. We propose a novel heterogeneous ensemble approach based on online dynamic ensemble selection that accurately interchanges between different types of base models in an ensemble to enhance its predictive performance in nonstationary environments. The approach is known as Heterogeneous Dynamic Ensemble Selection based on Accuracy and Diversity (HDES-AD) and makes use of models generated by different base learners to increase diversity to circumvent problems associated with existing dynamic ensemble classifiers that may experience loss of diversity due to the exclusion of base learners generated by different base algorithms. The algorithm is evaluated on artificial and real-world datasets with well-known online homogeneous online ensemble approaches such as DDD, AFWE, and OAUE. The results show that HDES-AD performed significantly better than the other three homogeneous online ensemble approaches in nonstationary environments.

Adaptive Multi-Level Search for Global Optimization: An Integrated Swarm Intelligence-Metamodelling Technique

Over the last decade, metaheuristic algorithms have emerged as a powerful paradigm for global optimization of multimodal functions formulated by nonlinear problems arising from various engineering subjects. However, numerical analyses of many complex engineering design problems may be performed using finite element method (FEM) or computational fluid dynamics (CFD), by which function evaluations of population-based algorithms are repetitively computed to seek a global optimum. It is noted that these simulations become computationally prohibitive for design optimization of complex structures. To efficiently and effectively address this class of problems, an adaptively integrated swarm intelligence-metamodelling (ASIM) technique enabling multi-level search and model management for the optimal solution is proposed in this paper. The developed technique comprises two steps: in the first step, a global-level exploration for near optimal solution is performed by adaptive swarm-intelligence algorithm, and in the second step, a local-level exploitation for the fine optimal solution is studied on adaptive metamodels, which are constructed by the multipoint approximation method (MAM). To demonstrate the superiority of the proposed technique over other methods, such as conventional MAM, particle swarm optimization, hybrid cuckoo search, and water cycle algorithm in terms of computational expense associated with solving complex optimization problems, one benchmark mathematical example and two real-world complex design problems are examined. In particular, the key factors responsible for the balance between exploration and exploitation are discussed as well.

A Surrogate-Assisted Multiswarm Optimization Algorithm for High-Dimensional Computationally Expensive Problems

This article presents a surrogate-assisted multiswarm optimization (SAMSO) algorithm for high-dimensional computationally expensive problems. The proposed algorithm includes two swarms: the first one uses the learner phase of teaching-learning-based optimization (TLBO) to enhance exploration and the second one uses the particle swarm optimization (PSO) for faster convergence. These two swarms can learn from each other. A dynamic swarm size adjustment scheme is proposed to control the evolutionary progress. Two coordinate systems are used to generate promising positions for the PSO in order to further enhance its search efficiency on different function landscapes. Moreover, a novel prescreening criterion is proposed to select promising individuals for exact function evaluations. Several commonly used benchmark functions with their dimensions varying from 30 to 200 are adopted to evaluate the proposed algorithm. The experimental results demonstrate the superiority of the proposed algorithm over three state-of-the-art algorithms.