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Effective constraint handling techniques are of great significance for evolutionary algorithms dealing with constrained optimization problems. To date, many constraint handling techniques, such as penalty function, superiority of feasible solutions, and -constraint, have been designed. However, different constraint handling techniques are usually suited to different problems, even the most appropriate technique changes along with the stages of the optimization process. Motivated by this phenomenon, we propose a voting-mechanism based ensemble framework, named VMCH, to integrate multiple constraint handling techniques for solving various constrained optimization problems. In this framework, each constraint handling technique acts as a voter, all voters vote for each pair of solutions, and the solution in each pair with the highest weighted votes is considered better. In addition, an adaptive strategy is developed to adjust the voter weights according to their historical voting performance. To investigate the performance of VMCH in improving existing algorithms, the proposed VMCH is embedded into the three best algorithms in the competition on constrained single objective real-parameter optimization at CEC 2018, namely MAgES, iLSHADE, and IUDE, to form three new algorithm versions, i.e., MAgES-VMCH, iLSHADE-VMCH, and IUDE-VMCH. They are compared with seven state-of-the-art peer algorithms. Extensive experiments are conducted on 57 real-world constrained optimization problems. The ranking results show that the new algorithm version MAgES-VMCH takes first place among the ten comparison algorithms. Moreover, all the new VMCH-enhanced versions of the three best algorithms are superior to their original versions. Therefore, the proposed VMCH framework can achieve competitive performance in solving constrained optimization problems.

In this study, we propose an efficient optimization algorithm that is a hybrid of the iterated greedy and simulated annealing algorithms (hereinafter, referred to as IGSA) to solve the flexible job shop scheduling problem with crane transportation processes (CFJSP). Two objectives are simultaneously considered, namely, the minimization of the maximum completion time and the energy consumptions during machine processing and crane transportation. Different from the methods in the literature, crane lift operations have been investigated for the first time to consider the processing time and energy consumptions involved during the crane lift process. The IGSA algorithm is then developed to solve the CFJSPs considered. In the proposed IGSA algorithm, first, each solution is represented by a 2-D vector, where one vector represents the scheduling sequence and the other vector shows the assignment of machines. Subsequently, an improved construction heuristic considering the problem features is proposed, which can decrease the number of replicated insertion positions for the destruction operations. Furthermore, to balance the exploration abilities and time complexity of the proposed algorithm, a problem-specific exploration heuristic is developed. Finally, a set of randomly generated instances based on realistic industrial processes is tested. Through comprehensive computational comparisons and statistical analyses, the highly effective performance of the proposed algorithm is favorably compared against several efficient algorithms.

Probabilistic load forecasting (PLF) has become necessary for power system operators to do efficient planning across power transmission and distribution systems. However, there are not many PLF models, and those that exist take a lot of computation time and are not efficient, especially in multiple loads. This paper proposes a novel algorithm for spatially correlated multiple loads wherein a global parameter is learned from state-space parameters of individual loads by an amalgamation of deep neural networks and state-space models. The proposed model employs complex pattern learning capabilities of recurrent neural networks and temporal pattern extraction of innovation state-space models. It is tested on GEFCom-14 and ISO-NE datasets, one with a single load and multiple loads. Different case studies are conducted to examine the involvement of temperature for load forecasting. It has been observed that in the case of multivariate loads, temperature variable doesn’t make much difference in PLF, but in the case of univariate loads, forecasting results are four-times better. The proposed method is highly interpretable and can be employed in areas where limited training data is available to the areas where colossal data is available. The proposed model has outperformed several benchmarks present in the literature on the same datasets.

Sorting solutions play a key role in using evolutionary algorithms (EAs) to solve many-objective optimization problems (MaOPs). Generally, different solution-sorting methods possess different advantages in dealing with distinct MaOPs. Focusing on this characteristic, this article proposes a general voting-mechanism-based ensemble framework (VMEF), where different solution-sorting methods can be integrated and work cooperatively to select promising solutions in a more robust manner. In addition, a strategy is designed to calculate the contribution of each solution-sorting method and then the total votes are adaptively allocated to different solution-sorting methods according to their contribution. Solution-sorting methods that make more contribution to the optimization process are rewarded with more votes and the solution-sorting methods with poor contribution will be punished in a period of time, which offers a good feedback to the optimization process. Finally, to test the performance of VMEF, extensive experiments are conducted in which VMEF is compared with five state-of-the-art peer many-objective EAs, including NSGA-III, SPEA/R, hpaEA, BiGE, and grid-based evolutionary algorithm. Experimental results demonstrate that the overall performance of VMEF is significantly better than that of these comparative algorithms.

This paper addresses a traffic signal scheduling (TSS) problem in a heterogeneous traffic network with signalized and non-signalized intersections. The objective is to minimize the total network-wise delay time of all vehicles within a given finite-time window. First, a novel model is proposed to describe a heterogeneous traffic network with signalized and non-signalized intersections. Second, five meta-heuristics are implemented to solve the TSS problem. Based on the problem characteristics, three local search operators and their ensemble are proposed. Then, five meta-heuristics with such an ensemble are proposed to solve the TSS problem. Third, experiments are carried out based on the real traffic data in the Jurong area of Singapore. The performance of the ensemble of local search operators is verified. Ten algorithms, including five meta-heuristics with and without the ensemble, are evaluated by solving 18 cases with different scales. Finally, the algorithm with the best performance is compared against the currently used traffic signal control strategies. The comparisons and discussions show the competitiveness of the proposed model and meta-heuristics.

This paper addresses a bi-objective urban traffic light scheduling problem (UTLSP), which requires minimizing both the total network-wise delay time of all vehicles and total delay time of all pedestrians within a given finite-time window. First, a centralized model is employed to describe the UTLSP, where the cost functions and constraints of the two objectives are presented. A non-domination strategy-based metric is used to compare and rank solutions based on the two objectives. Second, metaheuristics, such as harmony search (HS) and artificial bee colony (ABC), are implemented to solve the UTLSP. Based on the characteristics of the UTLSP, a local search operator is utilized to improve the search performance of the developed optimization algorithms. Finally, experiments are carried out based on the real traffic data in Jurong area of Singapore. The HS, ABC, and their variants with the local search operator are evaluated in 19 case studies with different scales and time windows. To the best of our knowledge, this paper is the first of its kind to solve bi-objective traffic light scheduling problems in the literature. To demonstrate the effectiveness of the proposed algorithms in dealing with bi-objective optimization in traffic light scheduling, they are compared to the classical non-dominated sorting genetic algorithm II (NSGAII) with and without the local search operation. The comparisons indicate that our algorithms outperform the NSGAII algorithm with and without the local search operator for solving the UTLSP.

In this letter, an efficient indicator for multi and many-objective optimization is proposed. The proposed indicator ( $I_{{SDE}}$ +) is a combination of sum of objectives and shift-based density estimation and benefits from their ability to promote convergence and diversity, respectively. An evolutionary multiobjective optimization framework based on the proposed indicator is shown to perform comparably or better than the state-of-the-art on a variety of scalable benchmark problems.

In evolutionary computation, balancing the diversity and convergence of the population for multiobjective evolutionary algorithms (MOEAs) is one of the most challenging topics. Decomposition-based MOEAs are efficient for population diversity, especially when the branch partitions the objective space of multiobjective optimization problem (MOP) into a series of subspaces, and each subspace retains a set of solutions. However, a persisting challenge is how to strengthen the population convergence while maintaining diversity for decomposition-based MOEAs. To address this issue, we first define a novel metric to measure the contributions of subspaces to the population convergence. Then, we develop an adaptive strategy that allocates computational resources to each subspace according to their contributions to the population. Based on the above two strategies, we design an objective space partition-based adaptive MOEA, called OPE-MOEA, to improve population convergence, while maintaining population diversity. Finally, 41 widely used MOP benchmarks are used to compare the performance of the proposed OPE-MOEA with other five representative algorithms. For the 41 MOP benchmarks, the OPE-MOEA significantly outperforms the five algorithms on 28 MOP benchmarks in terms of the metric hypervolume.

Studies in machine learning have shown promising classification performance of ensemble methods employing "perturb and combine" strategies. In particular, the classical random forest algorithm performs the best among 179 classifiers on 121 UCI datasets from different domains. Motivated by this observation, we extend our previous work on oblique decision tree ensemble. We also propose an efficient co-trained kernel ridge regression method. In addition, a random vector functional link network ensemble is also introduced. Our experiments show that our two oblique decision tree ensemble variants and the co-trained kernel ridge regression ensemble are the top three ranked methods among the 183 classifiers. The proposed random vector functional link network ensemble also outperforms all neural network based methods used in the experiments.

The flowshop sequence-dependent group scheduling problem (FSDGSP) with the production efficiency measures has been extensively studied due to its wide industrial applications. However, energy efficiency indicators are often ignored in the literature. This paper considers the FSDGSP to minimize makespan, total flow time and total energy consumption, simultaneously. After the problem-specific knowledge is extracted, a mixed-integer linear programming model and a critical path-based accelerated evaluation method are proposed. Since the FSDGSP includes multiple coupled sub-problems, a greedy cooperative co-evolutionary algorithm (GCCEA) is designed to explore the solution space in depth. Meanwhile, a random mutation operator and a greedy energy-saving strategy are employed to adjust the processing speeds of machines to obtain a potential non-dominated solution. A large number of experimental results show that the proposed algorithm significantly outperforms the existing classic multi-objective optimization algorithms, which is due to the usage of problem-related knowledge.

Many researchers have developed population-based techniques to solve numerical optimization problems. Almost none of these techniques demonstrate consistent performance over a wide range of problems as these problems differ substantially in their characteristics. In the state-of-the-art cultural algorithms (CAs), problem solving is facilitated by the exchange of knowledge between a network of active knowledge sources in the belief space and networks of individuals in the population space. To enhance the performance of CAs, we restructure the social fabric interconnections to facilitate flexible communication among problem solvers in the population space. Several social network reconfiguration mechanisms and types of communications are examined. This extended CA is compared with other variants of CAs and other well-known state-of-the-art algorithms on a set of challenging real-world problems. The numerical results show that the injection of neighborhoods with flexible subnetworks enhances performance on a diverse landscape of numerical optimization problems.

Ensemble methods use multiple models to get better performance. Ensemble methods have been used in multiple research fields such as computational intelligence, statistics and machine learning. This paper reviews traditional as well as state-of-the-art ensemble methods and thus can serve as an extensive summary for practitioners and beginners. The ensemble methods are categorized into conventional ensemble methods such as bagging, boosting and random forest, decomposition methods, negative correlation learning methods, multi-objective optimization based ensemble methods, fuzzy ensemble methods, multiple kernel learning ensemble methods and deep learning based ensemble methods. Variations, improvements and typical applications are discussed. Finally this paper gives some recommendations for future research directions.

Multimodal optimization problems consists of multiple equal or comparable spatially distributed solutions. Niching and clustering differential evolution (DE) techniques have been demonstrated to be highly effective for solving such problems. The key challenge in the speciation niching technique is to balance between local solution exploitation and global exploration. Our proposal enhances exploration by applying arithmetic recombination with speciation and improves exploitation of individual peaks by applying neighborhood mutation with ensemble strategies. Our novel algorithm, called ensemble and arithmetic recombination-based speciation DE, is shown to either outperform or perform comparably to the state-of-the-art algorithms on 29 common multimodal benchmark problems. Comparable performance is observed only when some problems are solved perfectly by the algorithms in the literature.

A new approach to generate oblique decision tree ensemble is proposed wherein each decision hyperplane in the internal node of tree classifier is not always orthogonal to a feature axis. All training samples in each internal node are grouped into two hyper-classes according to their geometric properties based on a randomly selected feature subset. Then multisurface proximal support vector machine is employed to obtain two clustering hyperplanes where each hyperplane is generated such that it is closest to one group of the data and as far as possible from the other group. Then, one of the bisectors of these two hyperplanes is regarded as the test hyperplane for this internal node. Several regularization methods have been applied to handle the small sample size problem as the tree grows. The effectiveness of the proposed method is demonstrated by 44 real-world benchmark classification data sets from various research fields. These classification results show the advantage of the proposed approach in both computation time and classification accuracy.

Wind speed forecasting is challenging due to its intermittent nature. The wind speed time series (TS) has nonlinear and nonstationary characteristics and not normally distributed, which make it difficult to be predicted by statistical or computational intelligent methods. Empirical mode decomposition (EMD) and its improved versions are powerful tools to decompose a complex TS into a collection of simpler ones. The improved versions discussed in this paper include ensemble EMD (EEMD), complementary EEMD (CEEMD), and complete EEMD with adaptive noise (CEEMDAN). The EMD and its improved versions are hybridized with two computational intelligence-based predictors: support vector regression (SVR) and artificial neural network (ANN). The EMD-based hybrid forecasting methods are evaluated with 12 wind speed TS. The performances of the hybrid methods are compared and discussed. It shows that EMD and its improved versions enhance the performance of SVR significantly but marginally on ANN, and among the EMD-based hybrid methods, the proposed CEEMDAN-SVR is the best method. Possible future works are also recommended for wind speed forecasting.

Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space and when needed, the current solution may be switched to a more suitable one while still maintaining the optimal system performance. Niching particle swarm optimizers (PSOs) have been widely used by the evolutionary computation community for solving real-parameter multimodal optimization problems. However, most of the existing PSO-based niching algorithms are difficult to use in practice because of their poor local search ability and requirement of prior knowledge to specify certain niching parameters. This paper has addressed these issues by proposing a distance-based locally informed particle swarm (LIPS) optimizer, which eliminates the need to specify any niching parameter and enhance the fine search ability of PSO. Instead of using the global best particle, LIPS uses several local bests to guide the search of each particle. LIPS can operate as a stable niching algorithm by using the information provided by its neighborhoods. The neighborhoods are estimated in terms of Euclidean distance. The algorithm is compared with a number of state-of-the-art evolutionary multimodal optimizers on 30 commonly used multimodal benchmark functions. The experimental results suggest that the proposed technique is able to provide statistically superior and more consistent performance over the existing niching algorithms on the test functions, without incurring any severe computational burdens.

In this paper, a neighborhood mutation strategy is proposed and integrated with various niching differential evolution (DE) algorithms to solve multimodal optimization problems. Although variants of DE are highly effective in locating a single global optimum, no DE variant performs competitively when solving multi-optima problems. In the proposed neighborhood based differential evolution, the mutation is performed within each Euclidean neighborhood. The neighborhood mutation is able to maintain the multiple optima found during the evolution and evolve toward the respective global/local optimum. To test the performance of the proposed neighborhood mutation DE, a total of 29 problem instances are used. The proposed algorithms are compared with a number of state-of-the-art multimodal optimization approaches and the experimental results suggest that although the idea of neighborhood mutation is simple, it is able to provide better and more consistent performance over the state-of-the-art multimodal algorithms. In addition, a comparative survey on niching algorithms and their applications are also presented.

Differential evolution (DE) is one of the most powerful stochastic real parameter optimizers of current interest. In this paper, we propose a new mutation strategy, a fitness-induced parent selection scheme for the binomial crossover of DE, and a simple but effective scheme of adapting two of its most important control parameters with an objective of achieving improved performance. The new mutation operator, which we call DE/current-to-gr_best/1, is a variant of the classical DE/current-to-best/1 scheme. It uses the best of a group (whose size is q% of the population size) of randomly selected solutions from current generation to perturb the parent (target) vector, unlike DE/current-to-best/1 that always picks the best vector of the entire population to perturb the target vector. In our modified framework of recombination, a biased parent selection scheme has been incorporated by letting each mutant undergo the usual binomial crossover with one of the p top-ranked individuals from the current population and not with the target vector with the same index as used in all variants of DE. A DE variant obtained by integrating the proposed mutation, crossover, and parameter adaptation strategies with the classical DE framework (developed in 1995) is compared with two classical and four state-of-the-art adaptive DE variants over 25 standard numerical benchmarks taken from the IEEE Congress on Evolutionary Computation 2005 competition and special session on real parameter optimization. Our comparative study indicates that the proposed schemes improve the performance of DE by a large magnitude such that it becomes capable of enjoying statistical superiority over the state-of-the-art DE variants for a wide variety of test problems. Finally, we experimentally demonstrate that, if one or more of our proposed strategies are integrated with existing powerful DE variants such as jDE and JADE, their performances can also be enhanced.

Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms in current use. DE operates through similar computational steps as employed by a standard evolutionary algorithm (EA). However, unlike traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used for generating the offspring. Since its inception in 1995, DE has drawn the attention of many researchers all over the world resulting in a lot of variants of the basic algorithm with improved performance. This paper presents a detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far. Also, it provides an overview of the significant engineering applications that have benefited from the powerful nature of DE.

An ecologically inspired optimization algorithm, called invasive weed optimization (IWO), is presented for the design of non-uniform, planar, and circular antenna arrays that can achieve minimum side lobe levels for a specific first null beamwidth while avoiding the mutual coupling effects simultaneously. IWO recently emerged as a derivative-free real parameter optimizer that mimics the ecological behavior of colonizing weeds. For the present application, classical IWO has been modified by introducing a more explorative routine of changing the standard deviation of the seed population (equivalent to mutation step-size in evolutionary algorithms) of the algorithm. Simulation results over three significant instances of the circular array design problem have been presented to illustrate the effectiveness of the modified IWO algorithm. The design results obtained with modified IWO have been shown to comfortably beat those obtained with other state-of-the-art metaheuristics like genetic algorithm (GA), particle swarm optimization (PSO), original IWO and differential evolution (DE) in a statistically meaningful way.

During the last three decades, several constraint handling techniques have been developed to be used with evolutionary algorithms (EAs). According to the no free lunch theorem, it is impossible for a single constraint handling technique to outperform all other techniques on every problem. In other words, depending on several factors such as the ratio between feasible search space and the whole search space, multimodality of the problem, the chosen EA, and global exploration/local exploitation stages of the search process, different constraint handling methods can be effective during different stages of the search process. Motivated by these observations, we propose an ensemble of constraint handling techniques (ECHT) to solve constrained real-parameter optimization problems, where each constraint handling method has its own population. A distinguishing feature of the ECHT is the usage of every function call by each population associated with each constraint handling technique. Being a general concept, the ECHT can be realized with any existing EA. In this paper, we present two instantiations of the ECHT using four constraint handling methods with the evolutionary programming and differential evolution as the EAs. Experimental results show that the performance of ECHT is better than each single constraint handling method used to form the ensemble with the respective EA, and competitive to the state-of-the-art algorithms.

Differential evolution (DE) is an efficient and powerful population-based stochastic search technique for solving optimization problems over continuous space, which has been widely applied in many scientific and engineering fields. However, the success of DE in solving a specific problem crucially depends on appropriately choosing trial vector generation strategies and their associated control parameter values. Employing a trial-and-error scheme to search for the most suitable strategy and its associated parameter settings requires high computational costs. Moreover, at different stages of evolution, different strategies coupled with different parameter settings may be required in order to achieve the best performance. In this paper, we propose a self-adaptive DE (SaDE) algorithm, in which both trial vector generation strategies and their associated control parameter values are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, a more suitable generation strategy along with its parameter settings can be determined adaptively to match different phases of the search process/evolution. The performance of the SaDE algorithm is extensively evaluated (using codes available from P. N. Suganthan) on a suite of 26 bound-constrained numerical optimization problems and compares favorably with the conventional DE and several state-of-the-art parameter adaptive DE variants.

This paper presents a variant of particle swarm optimizers (PSOs) that we call the comprehensive learning particle swarm optimizer (CLPSO), which uses a novel learning strategy whereby all other particles' historical best information is used to update a particle's velocity. This strategy enables the diversity of the swarm to be preserved to discourage premature convergence. Experiments were conducted (using codes available from http://www.ntu.edu.sg/home/epnsugan) on multimodal test functions such as Rosenbrock, Griewank, Rastrigin, Ackley, and Schwefel and composition functions both with and without coordinate rotation. The results demonstrate good performance of the CLPSO in solving multimodal problems when compared with eight other recent variants of the PSO.

In this paper, we propose a novel approach to generate the topology-preserving mapping of structural shapes using self-organizing maps (SOMs). The structural information of the geometrical shapes is captured by relational attribute vectors. These vectors are quantised using an SOM. Using this SOM, a histogram is generated for every shape. These histograms are treated as inputs to train another SOM which yields a topology-preserving mapping of the geometric shapes. By appropriately choosing the relational vectors, it is possible to generate a mapping that is invariant to some chosen transformations, such as rotation, translation, scale, affine, or perspective transformations. Experimental results using trademark objects are presented to demonstrate the performance of the proposed methodology.

The six papers in this special issue are representative of the current research trends in differential evolution.

We develop a multilayer overlapped self-organizing maps (SOM's) with limited structure adaptation capabilities, and associated learning scheme for labeled pattern classification applications. The learning algorithm consists of the standard unsupervised SOM learning of synaptic weights as well as the supervised learning vector quantization (LVQ) 2 learning. As higher layer SOM's overlap, the final classification is made by fusing the classifications of top-level overlapped SOM's. We obtained the best results ever reported for any SOM-based numerals classification system.

As the number of fiber Bragg gratings (FBGs) increases, the conventional peak detection method will be unsuitable to detect Bragg wavelengths of FBG sensors in a wavelength-division-multiplexed (WDM) network. To solve this problem while achieving a higher accuracy at reduced computation cost, a novel tree search dynamic multiswarm particle swarm optimizer (TS-DMS-PSO) is developed. The TS-DMS-PSO yields better results with less computation cost based on the simulations (codes are available from the second author)

The design of a low-dispersion fiber Bragg grating (FBG) with an optimal grating length using covariance matrix adapted evolution strategy (CMAES) is presented. A novel objective function formulation is proposed for the optimal grating length low-dispersion FBG design. The CMAES algorithm employs adaptive learning procedure to identify correlations among the design parameters. The design of a low-dispersion FBG filter with 25-GHz (or 0.2 nm in the 1550-nm band) bandwidth is considered. Simulation results, obtained using the codes available in public domain (the codes are available from the third author), show that the CMAES algorithm is more appropriate for the practical design of length optimized FBG-based filters when compared with the other optimization methods.

We propose a novel formulation of the objective function for the design of fiber Bragg grating (FBG)-based filters with respect to the given design specifications, instead of matching the desired magnitude and phase responses of the filter at each wavelength of the operating window that has commonly been used in previous works on FBG synthesis. The desired reflective spectrum and group delay characteristics of a filter are predefined using six design specifications. Particle swarm optimization (PSO) technique is employed here to find an optimum index modulation profile that meets the target design. To demonstrate the effectiveness of the PSO algorithm and the novel formulation of the objective function, an optimal design of a low-dispersion FBG-based filter with 0.2-nm bandwidth (or 25 GHz in the 1550-nm window) for three desired values of the maximum reflective power is presented.