The process of training feedforward neural networks (FFNNs) can benefit from an automated process where the best heuristic to train the network is sought out automatically by means of a highlevel probabilistic-based heuristic.This research introduces a novel population-based Bayesian hyper-heuristic (BHH) that is used to train feedforward neural networks (FFNNs).The performance of the BHH is compared to that of ten popular low-level heuristics, each with different search behaviours.The chosen heuristic pool consists of classic gradient-based heuristics as well as metaheuristics (MHs).The empirical process is executed on fourteen datasets consisting of classification and regression problems with varying characteristics.The BHH is shown to be able to train FFNNs well and provide an automated method for finding the best heuristic to train the FFNNs at various stages of the training process.

Abstract Dynamic constrained optimization problems (DCOPs) are optimization problems where both the problem landscape and the problem constraints change over time, or either the problem landscape or the constraints change. Although DCOPs represent the super-set of optimization problems, relatively little is understood about these problems due to the complexity added to the optimization process and few meta-heuristics exist for DCOPs. This paper proposes a co-evolutionary meta-heuristic framework to allow for easy integration of existing dynamic meta-heuristics developed to solve box-constrained dynamic optimization problems only, into the framework to produce new co-evolutionary versions of these dynamic meta-heuristics to solve various classes of DCOPs. The paper analyzes the performance of the resulting co-evolutionary versions of these existing dynamic meta-heuristics on a comprehensive set of DCOP benchmark problems, and shows that the performance of these dynamic co-evolutionary algorithms are the best performing among a number of evaluated meta-heuristics.

Many nature-inspired metaheuristics have been published, with claims of originality based on the metaphor that inspired the algorithm. Rarely is empirical evidence given to show algorithmic originality. In order to provide an easy and computationally cheap approach to characterise algorithm search behaviour, a suite of 20 behavioural characteristics is proposed. This behavioural characteristic suite allows for the search behaviour of an algorithm to be quantified without manual inspection. By doing so, behavioural novelty of any given algorithm may be determined by comparing the behavioural characteristics to those of well-known metaheuristics. To illustrate this use, and to evaluate whether metaheuristics are behaviourally distinct, a host of metaheuristics is run on various benchmark functions. To evaluate behavioural similarity across all problems, while acknowledging behaviour to be problem dependant, a novel method is proposed. In addition to this method, new behavioural characteristics are also proposed. The behavioural vectors generated for each benchmark function are clustered. The relationships and trends present in the different clusters are summarised by creating a pair-wise matrix for every metaheuristic pair, which tallies the number of times that the pair are found within the same cluster. The tallies are then analysed in order to make inference regarding the distinctness of any metaheuristic’s behaviours, across many different benchmark functions. The analysis finds that the range of unique search behaviours is small and that most metaheuristics share their behaviours with most other metaheuristics. The analysis also identifies both unique algorithms, as well as algorithms which have no unique behaviours.

Streamed data classification problems (SDCPs) require classifiers to not just find the optimal decision boundaries that describe the relationships within a data stream, but also to adapt to changes in the decision boundaries in real-time. The requirement is due to concept drift, i.e., incorrect classifications caused by decision boundaries changing over time. Changes include disappearing, appearing or shifting decision boundaries. This article proposes an online learning approach for feed forward neural networks (FFNNs) that meets the requirements of SDCPs. The approach uses regularisation to dynamically optimise the architecture, and quantum particle swarm optimisation (QPSO) to dynamically adjust the weights. The learning approach is applied to a FFNN, which uses rectified linear activation functions, to form a novel SDCP classifier. The classifier is empirically investigated on several SDCPs. Both weight decay (WD) and weight elimination (WE) are investigated as regularisers. Empirical results show that using QPSO with no regularisation causes the classifier to completely saturate. However, using QPSO with regularisation makes the classifier efficient at dynamically adapting both its architecture and weights as decision boundaries change. Furthermore, the results favour WE over WD as a regulariser for QPSO.

Abstract Portfolio optimization has been studied extensively by researchers in computer science and finance, with new and novel work frequently published. Traditional methods, such as quadratic programming, are not computationally effective for solving complex portfolio models. For example, portfolio models with constraints that introduce nonlinearity and non-convexity (such as boundary constraints and cardinality constraints) are NP-Hard. As a result, researchers often use meta-heuristic approaches to approximate optimal solutions in an efficient manner. This paper conducts a comprehensive review of over 140 papers that have applied evolutionary and swarm intelligence algorithms to the portfolio optimization problem. These papers are categorized by the type of portfolio optimization problem considered, i.e., unconstrained or constrained, and are further categorized by single-objective and multi-objective approaches. Furthermore, the various portfolio models used, as well as the constraints, objectives, and properties in which they differ, are also discussed in a detailed analysis. Based on the findings of the reviewed work, guidance for future research in portfolio optimization is given. Possible areas for future work include dynamic portfolio optimization, predictive pricing, the further investigation of multi-objective approaches.

This study conducts a sensitivity analysis of the recently proposed multi-guide particle swarm optimisation (MG-PSO) algorithm for dynamic multi-objective optimisation problems (DMOPs). The MGPSO is a multi-swarm approach where each subswarm optimises one of the objectives. This paper further adapts the MGPSO algorithm to solve DMOPs by proposing alternative balance coefficient update strategies to allow efficient tracking of the changing Pareto-optimal front (POF). A total of twenty-nine benchmark functions and six performance measures were implemented to help with this task. The experiments were run against five different environment types to determine whether the MGPSO can solve problems with various spatial and temporal severities. The best control parameter update strategy was then compared with other state-of-the-art dynamic multi-objective optimisation algorithms (DMOAs). An extensive empirical analysis shows that MGPSO with the balance coefficient parameter re-initialized after the environment change achieves very competitive and oftentimes better performance when compared with the competing algorithms.

This paper presents a new approach to induce a list of rules from a dataset by using a set-based particle swarm optimisation algorithm. Many contemporary rule induction algorithms tend to use similar information gain based approaches to fit a training dataset. The proposed novel algorithm is a meta-heuristic approach which finds an optimal rule list while providing the flexibility to overcome traditional drawbacks such as overfitting and rigidity to the datatypes that can be used. This paper shows that the proposed algorithm performs comparatively well when compared to existing rule induction algorithms and it has the potential to be expanded further by adding rule pruning techniques.

Fair algorithm evaluation is conditioned on the existence of high-quality benchmark datasets that are non-redundant and are representative of typical optimization scenarios. In this paper, we evaluate three heuristics for selecting diverse problem instances which should be involved in the comparison of optimization algorithms in order to ensure robust statistical algorithm performance analysis. The first approach employs clustering to identify similar groups of problem instances and subsequent sampling from each cluster to construct new benchmarks, while the other two approaches use graph algorithms for identifying dominating and maximal independent sets of nodes. We demonstrate the applicability of the proposed heuristics by performing a statistical performance analysis of five portfolios consisting of three optimization algorithms on five of the most commonly used optimization benchmarks.

Selecting appropriate control parameters for the particle swarm optimization algorithm can be extremely time consuming and expensive, yet it is necessary in order to achieve optimal performance on a problem. Despite its significance, the issue of control parameter selection remains an open problem. This work leverages techniques from the field of fitness landscape analysis to characterize a large suite of benchmark problems. Extensive experimentation is performed to identify strong control parameters for each problem, and machine learning techniques are used to predict strong control parameters from the characterization of a problem. The results demonstrate that good generalization is possible with minimal training data. This suggests that the cost of parameter selection can be significantly reduced.

When designing effective parameter tuning and/or self-adaptive mechanisms for meta-heuristic optimizers, any insights about the configuration process and its associated landscape are of great benefit. Recently, the parameter configuration landscape (PCL) was proposed as a mechanism to formally study and characterize the landscape induced by the control parameter values of meta-heuristic search techniques. As an extension, the use of geomorphon landform types to further characterize and visualize the PCL was recently proposed. This study adopts the geomorphon classification scheme and applies it to particle swarm optimization (PSO). The methodology is applied on 20 minimization benchmark problems with various problem dimensions and swarm sizes, thereby providing deep insights into the PCL associated with PSO.

Particle swarm optimization (PSO) has been adapted to solve multi-objective optimization problems. However, these PSO-based multi-objective optimization algorithms typically face difficulties when the number of decision variables is increased and the problems turn into large-scale multi-objective problems (LSMOPs). This paper presents a decision space scalability analysis of five PSO-based multi-objective optimization algorithms, namely optimized multi-objective particle swarm op-timization (OMOPSO), speed-constrained multi-objective particle swarm optimization (SMPSO), multi-objective particle swarm optimization with multiple search strategies (MMOPSO), multi-guide particle swarm optimization (MGPSO), and competitive mechanism-based multi-objective particle swarm optimization (CMOPSO) for 24, 50, 100, 500 and 1000 dimensions (decision variables) to see how well each one of the algorithms scales as the number of decision variables is increased. The results indicate that, with an increase in the number of decision variables, MMOPSO and SMPSO had the best scalability, each dominating specific functions. Moreover, despite MGPSO's competitive performance on the 24-dimensional functions, it showed the worst overall scalability together with CMOPSO.

A number of particle swarm optimization (PSO) variations have been developed to find multiple solutions to multimodal optimization problems. These algorithms have been extensively evaluated in the literature. When dynamic optimization problems are considered, only a few PSO algorithms exist that have the ability to find and track multiple optima in dynamically changing search landscapes. These algorithms have not yet been rigorously evaluated on an extensive set of dynamic optimization problems. This paper presents a review of existing dynamic multimodal PSO algorithms and conducts an empirical analysis of these algorithms on a set of dynamic optimization problems of varying dynamics. The best performing dynamic multi-modal PSO algorithms, with respect to different performance measures, are identified as an outcome of a formal statistical analysis.

Fitness landscape analysis is an approach used to mathematically characterize optimization problems. Random walk algorithms are used to sample fitness landscapes in order to perform fitness landscape analysis. Random walk algorithms have an advantage over random samples, in that random walk algorithms keep note of successive points in the walk, along with the relationships between them. It is important that the sample generated by a random walk algorithm is representative of the entire fitness landscape. A representative sample can be said to have good coverage of the decision space of the optimization problem. A new measure of the coverage of random walk algorithms, i.e. the Hausdorff distance, is proposed. The coverage of random walk algorithms found in the literature is investigated using the Hausdorff distance. This study shows that it is not sufficient to consider only the robustness of a random walk algorithm when performing fitness landscape analysis, but that the coverage of decision space should also be considered. This study shows that there is no significant difference in the coverage provided by the random walk algorithms investigated. However, the differences between the coverage of the random walk algorithms is more prominent when the length of the random walks is short, or the dimensionality of the optimization problem is increased.

A hyper-heuristic is an optimization approach that continually selects the most appropriate heuristic(s) to apply to an optimization problem. Hyper-heuristics conduct a search in the space of heuristics, or heuristic space, for the most suitable heuristic to apply to candidate solutions in problem space. Traditionally, hyper-heuristics manage relatively simple low-level heuristics, which are often based on human domain intuition. Increasingly, hyper-heuristics are being used in conjunction with population-based meta-heuristics as the low-level heuristics. A heuristic space diversity measure helps practitioners understand the behavior of hyper-heuristics that manage population-based heuristics. This paper discusses existing measures to quantity heuristic space diversity, highlights shortcomings of these existing measures, and proposes a new heuristic space diversity entropy-based measure. Spatial and temporal volatility measures that characterize entity-to-heuristic assignments are also proposed.

Fitness landscape analysis is used to mathematically characterize optimization problems. In order to perform fitness landscape analysis on continuous-valued optimization problems, a sample of the fitness landscape needs to be taken. A common way to perform this sampling is to use random walk algorithms. This paper proposes a new random walk algorithm for continuous-valued optimization problems, called the distributed random walk algorithm. The algorithm is based on the premise that multiple short random walks of the same type will provide better coverage of the decision space and more robust fitness landscape measures than a single long random walk. The distributed random walk algorithm is simple to implement, and the computational overhead is insignificant compared to random walk algorithms in the literature. The results of the study indicate that the distributed random walk algorithm achieves both of these objectives. Furthermore, the benefits of the distributed random walk algorithm are shown to be much more significant when small step sizes are used in the random walks.

It has been argued in the past that high-dimensional neural networks do not exhibit local minima capable of trapping an optimisation algorithm. However, the relationship between loss surface modality and the neural architecture parameters, such as the number of hidden neurons per layer and the number of hidden layers, remains poorly understood. This study employs fitness landscape analysis to study the modality of neural network loss surfaces under various feed-forward architecture settings. An increase in the problem dimensionality is shown to yield a more searchable and more exploitable loss surface. An increase in the hidden layer width is shown to effectively reduce the number of local minima, and simplify the shape of the global attractor. An increase in the architecture depth is shown to sharpen the global attractor, thus making it more exploitable.

The main focus of this thesis is to compare the ability of various swarm intelligence algorithms when applied to the training of artificial neural networks. In order to compare the performance of the selected swarm intelligence algorithms both classification and regression datasets were chosen from the UCI Machine Learning repository. Swarm intelligence algorithms are compared in terms of training loss, training accuracy, testing loss, testing accuracy, hidden unit saturation, and overfitting. Our observations showed that Particle Swarm Optimization (PSO) was the best performing algorithm in terms of Training loss and Training accuracy. However, it was also found that the performance of PSO dropped considerably when examining the testing loss and testing accuracy results. For the classification problems, it was found that firefly algorithm, ant colony optimization, and fish school search outperformed PSO for testing loss and testing accuracy. It was also observed that ant colony optimization was the algorithm that performed the best in terms of hidden unit saturation.

It is well known that tuning a meta-heuristic optimizer is a challenging, yet rewarding process. Despite the benefits of a properly tuned optimizer, there is very little that is understood about the actual tuning process - many automated parameter tuning methods use an assumption that parameter configurations near a promising configuration will also be promising. However, this assumption has not been verified, in general. While the field of fitness landscape analysis can provide insight into the difficulty of an optimization problem, these techniques have not yet been applied to the parameter tuning problem. This paper proposes a methodology to apply standard techniques from fitness landscape analysis to the parameter configuration landscape of an arbitrary optimizer. This allows the characterization of the parameter tuning problem for an arbitrary optimizer on an arbitrary optimization problem. The proposed methodology is then investigated for the particle swarm optimization (PSO) algorithm on 20 benchmark problems in both 10 and 30 dimensions. The results indicate that the parameter configuration landscape of the PSO algorithm is globally unimodal, yet not necessarily an easy landscape to search. Furthermore, it is found that the characteristics of the PSO parameter configuration landscape do not correlate with the characteristics of the target benchmark problems.

This paper conducts a sensitivity analysis of the recently proposed multi-objective optimizing algorithm, namely the multi-guide particle swarm optimization algorithm (MGPSO). The MGPSO uses subswarms to explore the search space, where each subswarm optimises one of the multiple objectives. A bounded archive is used to share previously found non-dominated solutions between subswarms. A third term, the archive guide, is added to the velocity update equation that represents a randomly selected solution from the archive. The influence of the archive guide on a particle is controlled by the archive balance coefficient and is proportional to the social guide. The original implementation of the MGPSO used static values randomly sampled from a uniform distribution in the range [0,1] for the archive balance coefficient. This paper investigates a number of approaches to dynamically adjust this control parameter. These approaches are evaluated on a variety of multi-objective optimization problems. It is shown that a linearly increasing strategy and stochastic strategies outperformed the standard approach to initializing the archive balance coefficient on two-objective and three objective optimization problems.

A deep Continuous Conditional Recurrent Neural Fields (CCRNF) framework is presented in this paper to model dimensional emotion from multiple input features. The deep architecture is effected by stacking multiple gated recurrent neural networks to model complex, non-linear relationships across time and space. The effect of increasing layer depth is studied through a comparative performance analysis and a visual depiction of the gate activations. The resulting visual analysis provides insight into the flow of information across time and multiple layers. The paper further investigates the use of model uncertainty as captured in the Gaussian distribution of the model, and explores the use of inverse variances in the fusion of model decisions. This latter study serves as an initial discussion in quantifying model and prediction confidence in continuous conditional random fields.

Over the past few decades, a plethora of computational intelligence algorithms designed to solve multiobjective problems have been proposed in the literature. Unfortunately, it has been shown that a large majority of these optimizers experience performance degradation when tasked with solving problems possessing more than three objectives, referred to as many-objective problems (MaOPs). The downfall of these optimizers is that simultaneously maintaining a uniformly-spread set of solutions along with appropriate selection pressure to converge toward the Pareto-optimal front becomes significantly difficult as the number of objectives increases. This difficulty is further compounded for large-scale MaOPs, i.e., MaOPs with a large number of decision variables. In this paper, insight is given into the current state of many-objective research by investigating scalability of state-of-the-art algorithms using 3-15 objectives and 30-1000 decision variables. Results indicate that evolutionary optimizers are generally the best performers when the number of decision variables is low, but are outperformed by the swarm intelligence optimizers in several large-scale MaOP instances. However, a recently proposed evolutionary algorithm which combines dominance and subregion-based decomposition is shown to be promising for handling the immense search spaces encountered in large-scale MaOPs.

Particle swarm optimization (PSO) is a stochastic search algorithm based on the social dynamics of a flock of birds. The performance of the PSO algorithm is known to be sensitive to the values assigned to its control parameters. While many studies have provided reasonable ranges in which to initialize the parameters based on their long-term behaviours, such previous studies fail to quantify the empirical performance of parameter configurations across a wide variety of benchmark problems. This paper specifically address this issue by examining the performance of a set of 1012 parameter configurations of the PSO algorithm over a set of 22 benchmark problems using both the global-best and local-best topologies. Results indicate that, in general, parameter configurations which are within close proximity to the boundaries of the best-known theoretically-defined convergent region lead to better performance than configurations which are further away. Moreover, results indicate that neighbourhood topology plays a far more significant role than modality and separability when determining the regions in parameter space which perform well.

Multi-objective optimization (MOO) algorithms often use external archives to keep track of the Pareto-optimal solutions. Vector evaluated particle swarm optimization (VEPSO) is one such algorithm. In contrast to other MOO algorithms, VEPSO does not clearly define how to implement the archive. In this paper, the performance of various archive implementations, as found throughout the literature, are evaluated using the well-known Inverted Generational Distance (IGD) measure. A new archive implementation based on the hypersurface contribution is proposed and evaluated. The results show that overall the well-known crowding distance archive outperformed all other archive implementations. The hypersurface contribution archive also showed promise. Finally, it is shown that the distance metric and nearest neighbor archives perform worse than even the random archive.

At present the fitness-distance-ratio particle swarm optimizer (FDR-PSO) has undergone no form of theoretical stability analysis. This paper theoretically derives the conditions necessary for order-1 and order-2 stability under the well known stagnation assumption. Since it has been shown that particle stability has a meaningful impact on PSO's performance, it is important for PSO practitioners to know the actual criteria for particle stability. This paper validates its theoretical findings against an assumption free FDR-PSO algorithm. This empirical validation is necessary for a truly accurate representation of FDR-PSO's stability criteria.

Multimodal optimization (MMO) aiming to locate multiple optimal (or near-optimal) solutions in a single simulation run has practical relevance to problem solving across many fields. Population-based meta-heuristics have been shown particularly effective in solving MMO problems, if equipped with specifically-designed diversity-preserving mechanisms, commonly known as niching methods. This paper provides an updated survey on niching methods. This paper first revisits the fundamental concepts about niching and its most representative schemes, then reviews the most recent development of niching methods, including novel and hybrid methods, performance measures, and benchmarks for their assessment. Furthermore, this paper surveys previous attempts at leveraging the capabilities of niching to facilitate various optimization tasks (e.g., multiobjective and dynamic optimization) and machine learning tasks (e.g., clustering, feature selection, and learning ensembles). A list of successful applications of niching methods to real-world problems is presented to demonstrate the capabilities of niching methods in providing solutions that are difficult for other optimization methods to offer. The significant practical value of niching methods is clearly exemplified through these applications. Finally, this paper poses challenges and research questions on niching that are yet to be appropriately addressed. Providing answers to these questions is crucial before we can bring more fruitful benefits of niching to real-world problem solving.

Gramophone records were the main recording medium for more than seven decades and regained widespread popularity over the past several years. Being an analog storage medium, gramophone records are subject to distortions caused by scratches, dust particles, degradation, and other means of improper handling. The observed noise often leads to an unpleasant listening experience and requires a filtering process to remove the unwanted disruptions and improve the audio quality. This paper proposes a novel approach that employs various feed forward time delay artificial neural networks to detect and reconstruct noise in musical sound waves. A set of 800 songs from eight different genres were used to validate the performance of the neural networks. The performance was analyzed according to the outlier detection and interpolation accuracy, the computational time and the tradeoff between the accuracy and the time. The empirical results of both detection and reconstruction neural networks were compared to a number of other algorithms, including various statistical measurements, duplication approaches, trigonometric processes, polynomials, and time series models. It was found that the neural networks' outlier detection accuracy was slightly lower than some of the other noise identification algorithms, but achieved a more efficient tradeoff by detecting most of the noise in real time. The reconstruction process favored neural networks with an increase in the interpolation accuracy compared to other widely used time series models. It was also found that certain genres such as classical, country, and jazz music were interpolated more accurately. Volatile signals, such as electronic, metal, and pop music were more challenging to reconstruct and were substantially better interpolated using neural networks than the other examined algorithms.

At present, very little theoretical analysis has been performed on the unified particle swarm optimizer (UPSO). This paper derives the order-1 and order-2 stable regions for the UPSO algorithm, along with the fixed point of particle convergence. The impact that the unification factor has on the stability of UPSO is also analyzed. The theoretical analysis is performed under the stagnation assumption; however, the derived results are shown to be both necessary and sufficient for particle convergence empirically, using a standardized methodology for assumption free convergence region analysis.

The performance of the Particle Swarm Optimization (PSO) algorithm can be greatly improved if the parameters are appropriately tuned. However, tuning the control parameters for PSO algorithms has traditionally been a time-consuming, empirical process. Furthermore, ideal parameters may be time-dependent. To address the issue of parameter tuning, self-adaptive PSO (SAPSO) algorithms adapt the PSO control parameters over time. While many such SAPSO techniques have been proposed, their behaviour is not well understood as no in-depth critical analysis of their adaptation mechanisms has been performed. This study examines the convergence behaviour of eight SAPSO algorithms both analytically and empirically. Evidence clearly indicates that the field of self-adaptive PSO algorithms is in a sad state, given that many techniques either demonstrate divergent behaviour coupled with excessive invalid particles, and thus infeasible solutions, or have prohibitively low particle step sizes caused by rapid convergence.

The quantum particle swarm optimization (QPSO) algorithm is a variant of the traditional particle swarm optimization (PSO) algorithm aimed at solving dynamic optimization problems. Some particles in the QPSO algorithm are selected as “quantum” particles and the positions of these particles are sampled, using some probability distribution, within a radius (i.e., a hypersphere) around the global best position while the remainder of particles follow standard PSO behaviour. The exploration and exploitation of the QPSO algorithm is heavily influenced by the probability distribution used as well as the size of the quantum radius. However, the best probability distribution and radius size are both problem and environment dependent. This work proposes using a parent centric crossover (PCX) operator to generate the positions of quantum particles, thereby removing the need for radius and probability distribution parameters completely. Two variants are proposed and results indicate that both variants are superior to QPSO, especially in environments exhibiting high temporal severity.

Many real-world problems have more than one objective and are dynamic in nature, where either an objective function or constraint can vary over time. These problems are referred to as dynamic multi-objective optimisation problems (DMOOPs). A key challenge for dynamic multi-objective optimisation (DMOO) research is efficiently evaluating and analysing the performance of DMOO algorithms (DMOAs). This includes benchmarks, performance measures and the approach used to analyse the obtained results. Most research in recent years focussed on either dynamic single-objective or static multi-objective optimisation. In the field of DMOO, research focussed on unconstrained DMOOPs. A few papers have recently proposed constrained DMOOPs. Therefore, a key sub-challenge in DMOO is to have a standard benchmark suite that contains both unconstrained and constrained DMOOPs with various characteristics. In addition, the constraints used in the benchmarks should be guided by constraints that occur in real-world problems. Most approaches used to analyse the performance of DMOAs do not take into account how well a DMOA tracks the changing optimal solutions over time, i.e. how well it performs in each of the various environments. Furthermore, there are still certain DMOOPs that the proposed algorithms struggle to solve. Therefore, more research is required with regards to the development of algorithms that can solve DMOOPs efficiently. Another important aspect of DMOO is the decision making process that can either occur offline or interactively. This paper discusses these key challenges and progress that has been made to address these challenges. Furthermore, actions to deal with the outstanding issues are also proposed.

Many real-world optimization problems contain multiple (often conflicting) goals to be optimized simultaneously, commonly referred to as multi-objective problems (MOPs). Currently, there exists a plethora of Pareto optimizers designed to solve MOPs. Previous literature has demonstrated that the performance of these optimizers degrade for problems which possess more than three objectives, known as many-objective problems (MaOPs). The downfall of the traditional Pareto approach is that the dominance-based selection strategy loses effectiveness in distinguishing desirable solutions as the number of objectives grows larger, inhibiting convergence to the true Pareto front. One potential solution to this problem is to utilize the concept of knee points as a secondary metric for optimization. Two new knee-driven algorithms are proposed within this work, namely the knee point driven particle swarm optimization (KnPSO) and knee point driven differential evolution (KnDE) algorithm. Due to the nature of the knee point identification mechanism used, both of these algorithms have the benefit of naturally producing a diverse set of solutions without having to incorporate additional criterion. The existing knee-driven evolutionary algorithm (KnEA) along with the proposed approaches are compared against several non-knee variants. Experimental results on nine challenging MaOPs demonstrate that knee points are a viable option for improving the performance of Pareto-based approaches. The knee point driven algorithms are shown to produce significantly higher inverted generational distance and hypervolume metric values in comparison to their non-knee counterparts.

Numerous niching algorithms have been developed to find multiple optima for multimodal optimisation problems. Analysis of niching algorithms presents a problem due to the multiple objectives of niching algorithms, that is, to find as many optima as possible, to find accurate solutions, and to find a diverse set of solutions. Performance analysis becomes more difficult when no knowledge of the fitness landscape is available. This is due to the assumption of most performance measures that information about the optima is available. This paper provides a critical review of existing performance measures for niching algorithms. In addition, the paper introduces a new approach, the mid-point technique, to determine a unique set of solutions for performance measurement. The performance measures are used to evaluate the performance of seven particle swarm optimisation niching algorithms. The obtained results are then used to compute the predictive capability of non-knowledge assuming performance measures to that of knowledge assuming measures. The analysis shows that the considered performance measures do show a convergent behaviour.

Artificial neural networks are inherently high-dimensional, which limits our ability to visualise and understand their inner workings. Neural network architecture and training algorithm parameters are usually optimised on an ad hoc basis, with very limited insight into the nature of the objective function landscape. This study proposes using fitness landscape analysis to quantify topological properties of neural network error landscapes. Five techniques from the fitness landscape analysis field are adapted to work with neural network error landscapes. These techniques are then used to analyse how the error landscape changes under different error measurements and different number of hidden layers. The results show that fitness landscape analysis provides valuable insight into neural network error landscapes, and could be used for architecture selection.

An explorative analysis in low dimensional objective space of the vector evaluated particle swarm optimization (VEPSO) algorithm is presented. Results indicate that the VEPSO algorithm continues to explore, and does not focus enough on exploitation. The resulting Pareto optimal fronts (POFs) have a poor spread and in some cases provide a poor estimate of the real POFs. It is hypothesized that the poor results can be attributed to the fact that the VEPSO algorithm is not exploiting enough. The multi guided VEPSO with random archive selection (MGVEPSOa) is introduced to address the lack of exploitation of the VEPSO algorithm. Results indicate that the MGVEPSOa algorithm outperforms the VEPSO algorithm leading to better POFs. Analysis of the candidate solutions of the MGVEPSOaalgorithm confirm that the algorithm exploits existing solutions more than the VEPSO algorithm. The improved performance is attributed to the increased exploitation confirming the hypothesis that VEPSO does not exploit enough.

A quantitative analysis in low dimensional objective space of the exploration behavior of the vector evaluated particle swarm optimization (VEPSO) algorithm is presented. A previous study showed that the VEPSO algorithm continues to explore the objective space, and does not focus enough on exploitation. To improve exploitation, the multi guided VEPSO with random archive selection was introduced. In this paper a new quantitive measurement, that tracks the particles' movement diversity in decision space, is developed. The results reinforce the conclusions drawn in earlier research. Additionally, the movement diversity measurement provides additional insight into why one of the two MGVEPSOaswarms continue to explore more of the objective space when tested on the problems in the Zitzler, Deb and Thiele (ZDT) test set.

Previous studies of feedforward neural networks (FFNNs) have found that asymptotically bounded activation functions used by particle swarm optimised (PSO) FFNNs have a significant impact on the swarm behaviour and FFNN performance. A number of alternative activation functions have however been developed that offer potential advantages over popularly used functions. The purpose of this study is to compare the Elliot, rectified linear, leaky rectified linear and softplus functions with the sigmoid and hyperbolic tangent functions on classification and regression problems. It is shown that the rectified linear function has equal performance to the sigmoid and hyperbolic tangent without the disadvantages of the bounded activation functions. Adaptive versions of the functions are compared on unscaled data sets using the PSO lambda-gamma algorithm. It is shown that shallower gradients are beneficial to accuracy, but that the FFNNs have inferior generalisation capability when compared to networks trained on scaled data.

This paper examines a group-based approach to stochastically scaling the cognitive and social components of a particle swarm's velocities. Usually, such scaling is done by generating a random vector and then multiplying in a component-wise fashion. Instead, the problem's decision variables are divided into a number of groups and every group is scaled with a random number. Three different grouping strategies are provided: fixed group number, decreasing group number and increasing group number. These grouping strategies are compared with a standard PSO and amongst one another on a well known suite of high-dimensional benchmark functions. The proposed grouping strategies were all more effective or equal to scaling the velocity components in the standard way. It was found that linearly increasing the number of groups that the decision variables are divided into significantly outperforms the standard update method and all of the other grouping strategies considered. A detailed discussion of the results obtained and a brief empirical investigation of any parameters introduced by the grouping strategies are also provided.

“Happy New Year!” At the beginning of 2016, I would like to take this opportunity to wish everyone a very happy, healthy, and prosperous new year! It is my great honor and privilege to serve as the Editor-in-Chief (EiC) of the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS (TNNLS), and I am excited to write this Editorial to start a new journey with you all.

The recognition of continuous dimensional emotion remains a challenging task due to large variations in the expression of emotion, and the difficulty of modeling emotion as temporal processes. This work proposes the use of a Nonlinear AutoRegressive with eXogenous inputs recurrent neural network (NARX-RNN) to learn emotional patterns in a given a dataset. The application of particle swarm optimisation in training the NARX-RNN is considered and compared to a gradient descent algorithm. We show that the NARX-RNN outperforms other methods in its emotion recognition ability, and can be easily trained with both gradient-free and gradient-based optimization methods.

This work provides a landscape analysis of the dynamic benchmark functions commonly used in multi-modal optimization. The benchmark analysis results reveal that the mechanisms responsible for dynamism in the current dynamic benchmarks do not significantly affect landscape features; thus suggesting a lack of representation for problems whose landscape features vary over time.

At present, the explicit conditions necessary for order-2 stability of the fully informed particle swarm optimizer (FIPS) have not be derived. This paper theoretically derives the criteria for order-2 stability of the FIPS algorithm under the stagnation assumption. The exact relationship between the criteria for order-2 stability and the neighborhood size is presented. The maximum possible convergence region is also presented for an arbitrarily large neighborhood size. Unlike the vast body of theoretical research on particle swarms, this paper validates its conclusions empirically against an assumption free FIPS algorithm. This empirical validation is necessary for a truly accurate representation of FIPS's convergence criteria.

Many real-world optimization problems contain multiple goals to be optimized concurrently. Vector-evaluated particle swarm optimization is a particle swarm optimization variant which employs multiple swarms to solve multi-objective optimization problems. Each swarm optimizes a single objective and information regarding current best positions is passed among swarms using a knowledge transfer strategy. This paper investigates the application of a local search technique to the vector-evaluated particle swarm optimization algorithm. A hill climbing algorithm is applied to non-dominated solutions, dominated solutions, swarm personal best positions and swarm global best positions. Performance of each local search strategy is compared with the standard vector-evaluated particle swarm optimization algorithm using various knowledge transfer strategies. The results indicate that three out of the four local search techniques significantly improved performance of the vector-evaluated particle swarm optimization algorithm for problems possessing two objectives. No significant performance improvement was found for three-objective problems.

Many real-world problems consist of multiple objectives that are in conflict with one another and dynamic in nature. These kinds of problems do not have a single solution, but a set of optimal trade-off solutions. These trade-off solutions are stored in a fixed-size archive during the optimisation process. A decision maker then decides which trade-off solution to use for a specific optimisation problem. Larger archives require more computations than smaller archives. However, no research has been conducted to determine the influence of the archive size on the performance of algorithms when solving these kinds of problems. Therefore, this paper investigates the effect of archive sizes on the performance of the dynamic vector evaluated particle swarm optimisation algorithm. In addition, this study investigates the effect of the sampling size of the true Pareto-optimal front (POF) on the performance measure values when using small archives. The results indicate that the archive size does influence the performance of the algorithm and that in certain cases a small archive size may be beneficial. Furthermore, the results indicate that a larger sampling size of the true POF results in a worse performance measure value for the smaller archives.

Real-world optimisation problems are usually constrained in some way. These constraints essentially modify the search space and can have a significant impact on the success of algorithms during optimisation. This paper proposes the notion of a violation landscape as a concept for analysing the nature of constrained continuous search spaces. A number of numerical measures are proposed for characterising constrained problems and these are tested on the CEC 2010 benchmark suite of constrained real-parameter optimisation problems. It is shown that for many constrained problems and algorithms, the features of the violation landscape are more relevant in terms of understanding algorithm performance than the features of the fitness landscape.

Particle swarm optimisation has been successfully applied as a neural network training algorithm before, often outperforming traditional gradient-based approaches. However, recent studies have shown that particle swarm optimisation does not scale very well, and performs poorly on high-dimensional neural network architectures. This paper hypothesises that hidden layer saturation is a significant factor contributing to the poor training performance of the particle swarms, hindering good performance on neural networks regardless of the architecture size. A selection of classification problems is used to test this hypothesis. It is discovered that although a certain degree of saturation is necessary for successful training, higher degrees of saturation ultimately lead to poor generalisation. Possible factors leading to saturation are suggested, and means of alleviating saturation in particle swarms through weight initialisation range, maximum velocity, and search space boundaries are analysed. This paper is intended as a preface to a more in-depth study of the problem of saturation in particle swarm optimisation as a neural network training algorithm.

Optimisation methods designed for static environments do not perform as well on dynamic optimisation problems as purpose-built methods do. Hyper-heuristics show great promise in handling dynamic environment dynamics because hyper-heuristics adapt to their environment. Different classifications of dynamic environments describe change dynamics such as spatial change severity, temporal change severity, homogeneity of peak movement, etc. Previous studies show that different hyper-heuristic selection mechanisms perform differently across different types of dynamic environments. This study investigates three hyper-heuristic selection methods with different selection pressures and shows an inverse correlation with environment change severity.

Several algorithms aimed at dynamic optimisation problems have been developed. This paper reports on the incorporation of a self-adaptive Brownian radius into competitive differential evolution (CDE). Four variations of a novel technique to achieving the self-adaptation is suggested and motivated. An experimental investigation over a large number of benchmark instances is used to determine the most effective of the four variations. The new algorithm is compared to its base algorithm on an extensive set of benchmark problems and its performance analysed. Finally, the new algorithm is compared to other algorithms by means of reported results found in the literature. The results indicate that CDE is improved the the incorporation of the self-adaptive Brownian radius and that the new algorithm compares well with other algorithms.

Many problems in the real-world have more than one objective, with at least two objectives in conflict with one another. In addition, at least one objective changes over time. These kinds of problems are called dynamic multi-objective optimisation problems (DMOOPs). Studies have shown that both the quantum particle swarm optimisation (QPSO) and charged particle swarm optimisation (CPSO) algorithms perform well in dynamic environments, since they maintain swarm diversity. Therefore, this paper investigates the effect of using either QPSOs or CPSOs in the sub-swarms of the dynamic vector-evaluated particle swarm optimisation (DVEPSO) algorithm. These DVEPSO variations are then compared against the default DVEPSO algorithm that uses gbest PSOs and DVEPSO using heterogeneous PSOs that contain both charged and quantum particles. Furthermore, all of the aforementioned DVEPSO configurations are compared against the dynamic multi-objective optimisation (DMOPSO) algorithm that was the winning algorithm of a comprehensive comparative study of dynamic multi-objective optimisation algorithms. The results indicate that charged and quantum particles improve the performance of DVEPSO, especially for DMOOPs with a deceptive POF and DMOOPs with a non-linear POS.

Most sensitivity analysis studies of optimization algorithm control parameters are restricted to a single objective function evaluation (OFE) budget. This restriction is problematic because the optimality of control parameter values (CPVs) is dependent not only on the problem's fitness landscape, but also on the OFE budget available to explore that landscape. Therefore, the OFE budget needs to be taken into consideration when performing control parameter tuning. This paper presents a new algorithm tuning multiobjective particle swarm optimization (tMOPSO) for tuning the CPVs of stochastic optimization algorithms under a range of OFE budget constraints. Specifically, for a given problem tMOPSO aims to determine multiple groups of CPVs, each of which results in optimal performance at a different OFE budget. To achieve this, the control parameter tuning problem is formulated as a multiobjective optimization problem. Additionally, tMOPSO uses a noise-handling strategy and CPV assessment procedure, which are specialized for tuning stochastic optimization algorithms. Conducted numerical experiments provide evidence that tMOPSO is effective at tuning under multiple OFE budget constraints.

This paper performs a thorough empirical investigation of the conditions placed on particle swarm optimization control parameters to ensure convergent behavior. At present there exists a large number of theoretically derived parameter regions that will ensure particle convergence, however, selecting which region to utilize in practice is not obvious. The empirical study is carried out over a region slightly larger than that needed to contain all the relevant theoretically derived regions. It was found that there is a very strong correlation between one of the theoretically derived regions and the empirical evidence. It was also found that parameters near the edge of the theoretically derived region converge at a very slow rate, after an initial population explosion. Particle convergence is so slow, that in practice, the edge parameter settings should not really be considered useful as convergent parameter settings.

New and existing optimisation algorithms are often compared by evaluating their performance on a benchmark suite. This set of functions aims to evaluate the algorithm across a range of problems and serves as a baseline measurement of how the algorithm may perform on real-world problems. It is important that the functions serve as a good representative of commonly occurring problems. In order to select functions that will make up the benchmark suite, the characteristics and relationships among the functions must be known. This paper characterises the landscapes of two commonly used benchmark suites, and uses these landscape characteristics to obtain a high level view of the current state of benchmark functions. This is done by using a self-organising feature map to cluster and analyse functions based on landscape characteristics. It is found that while there are numerous functions that cover a wide range of characteristics, there are characteristics that are under represented, or not even covered at all. Furthermore, it is discovered that common benchmark suites are composed of functions which are highly similar according to the measured characteristics.

Temporal data is common in real-world datasets. Clustering of such data allows for relationships between data patterns over time to be discovered. Differential evolution (DE) algorithms have previously been used to cluster temporal data. This paper proposes the cooperative data clustering dynamic DE algorithm (CDCDynDE), which is an adaptation to the data clustering dynamic DE (DCDynDE) algorithm where each population searches for a single cluster centroid. The paper applies the proposed algorithm to a variety of temporal datasets with different frequencies of change, severities of change, dataset dimensions and data migration types. The clustering results of the cooperative data clustering DynDE are compared against the original data clustering DynDE, the re-initialising data clustering DE and the standard data clustering DE. A statistical analysis of these results shows that the cooperative data clustering DynDE algorithm obtains better data clustering solutions to the other three algorithms despite changes in frequency, severity, dimension and data migration types.

This paper introduces the concept of heuristic space diversity and investigates various strategies for the management of heuristic space diversity within the context of a meta-hyper-heuristic algorithm. Evaluation on a diverse set of floating-point benchmark problems show that heuristic space diversity has a significant impact on hyper-heuristic performance. The increasing heuristic space diversity strategies performed the best out of all strategies tested. Good performance was also demonstrated with respect to another popular multi-method algorithm and the best performing constituent algorithm.

The vector evaluated particle swarm optimization (VEPSO) algorithm is a multi-swarm variation of the traditional particle swarm optimization (PSO) used to solve static multi-objective optimization problems (MOOPs). Recently, the dynamic VEPSO (DVEPSO) algorithm was proposed as an extension to VEPSO enabling the algorithm to handle dynamic MOOPs (DMOOPs). While DVEPSO has been successful at handling DMOOPs, the change detection mechanism relied on observing changes in objective space. An alternative strategy is proposed by using charged PSO (CPSO) sub-swarms with decision space change detection to address the outdated memory issue observed in vanilla PSO. This dynamic PSO variant allows for (implicit) decision space tracking not seen in DVEPSO while implicitly handling the diversity issue seen in dynamic environments. The proposed charged VEPSO is compared to DVEPSO on a wide variety of dynamic environment types. Results indicated that, in general, the proposed charged VEPSO outperformed the existing DVEPSO. Further, charged VEPSO exhibited better front-tracking abilities, while DVEPSO was superior with regards to locating the Pareto front.

Optimisation problems with more than one objective, where at least one objective changes over time, are called dynamic multi-objective optimisation problems (DMOOPs). Since at least two objectives are in conflict with one another, a single solution does not exist, and therefore the goal of a dynamic multi-objective optimisation algorithm (DMOA) is to track the set of optimal trade-off solutions over time. One of the major issues when solving optimisation problems, is balancing exploration and exploitation during the search process. This paper investigates the performance of the dynamic vector evaluated particle swarm optimisation (DVEPSO) algorithm using heterogeneous PSOs (HPSOs), where each particle has a different behaviour. The goal of the study is to determine whether the use of heterogeneous particle swarm optimisation (HPSO) algorithms will improve the performance of DVEPSO by incorporating particles with exploration and exploitation behaviour in a single particle swarm optimisation (PSO) algorithm. The results indicate that using HPSOs improves the performance of DVEPSO, especially for type I and type III DMOOPs.

F-Race and its variant, Iterated F-Race, is an automated procedure for sampling and evaluating potential values of parameters for algorithms. The procedure is controlled by means of a computational budget that limits the number of evaluations that may be conducted, thus forcing the determination of the best possible configuration to be made within a limited time. When time is not severely constrained, the a priori choice of a computational budget becomes unjustifiable because the relationship between the computational budget and the quality of the optimization of a black box subject is not obvious. This paper proposes an extension to F-Race in the form of a heuristic method for reasonably terminating the optimization procedure.

A number of fitness landscape analysis approaches are based on random walks through discrete search spaces. Applying these approaches to real-encoded problems requires the notion of a random walk in continuous space. This paper proposes a progressive random walk algorithm and the use of multiple walks to sample neighbourhood structure in continuous multi-dimensional spaces. It is shown that better coverage of a search space is provided by progressive random walks than simple unbiased random walks.

Portfolio optimization has as its objective to find optimal portfolios, which apportion capital between their constituent assets such that the portfolio's risk adjusted return is maximized. Portfolio optimization becomes more complex as constraints are imposed, multiple sources of return are included, and alternative measures of risk are used. Meta-heuristic portfolio optimization can be used as an alternative to deterministic approaches under increased complexity conditions. This paper uses a particle swarm optimization (PSO) algorithm to optimize a diversified portfolio of carry trades. In a carry trade, investors profit by borrowing low interest rate currencies and lending high interest rate currencies, thereby generating return through the interest rate differential. However, carry trades are risky because of their exposure to foreign exchange losses. Previous studies showed that diversification does significantly mitigate this risk. This paper goes one step further and shows that meta-heuristic portfolio optimization can further improve the risk adjusted returns of diversified carry trade portfolios.

A new competitive coevolutionary team-based particle swarm optimisation (CCPSO) algorithm is developed to train multi-agent teams from zero knowledge. The CCPSO algorithm uses the charged particle swarm optimiser to train neural network controllers for simple soccer agents. The training performance of the CCPSO algorithm is analysed. The analysis identifies a critical weakness of the CCPSO algorithm in the form of outliers in the measured performance of the trained players. A hypothesis is presented that explains the presence of the outliers, followed by a detailed discussion of various biased and unbiased relative fitness functions. A new relative fitness function based on FIFA's league ranking system is presented. The performance of the unbiased relative fitness functions is evaluated and discussed. The final results show that the FIFA league ranking relative fitness function outperforms the other unbiased relative fitness functions, leading to consistent training results.

This paper analyses the behaviour of particle swarm optimisation applied to training high-dimensional neural networks. Despite being an established neural network training algorithm, particle swarm optimisation falls short at training high-dimensional neural networks. Reasons for poor performance of PSO are investigated in this paper, and hidden unit saturation is hypothesised to be a cause of the failure of PSO in training high-dimensional neural networks. An analysis of various activation functions and search space boundaries leads to the conclusion that hidden unit saturation can be slowed down by combining activation function choice with appropriate search space boundaries. Bounded search is shown to significantly outperform unbounded search in high-dimensional neural network error search spaces.

Recent work has evaluated the performance of a synchronous global best (gbest) particle swarm optimization (PSO) algorithm hybridized with discrete crossover operators. This paper investigates if using asynchronous position updates instead of synchronous updates will result in improved performance of a gbest PSO that uses these discrete crossover operators. Empirical analysis of the performance of the resulting algorithms provides strong evidence that asynchronous position updates significantly improves performance of the PSO discrete crossover hybrid algorithms, mainly with respect to accuracy and convergence speed. These improvements were seen over an extensive benchmark suite of 60 boundary constrained minimization problems of various characteristics.

It has become acceptable practice to use only a limit on the number of fitness function evaluations (FEs) as a stopping condition when comparing population-based optimization algorithms, irrespective of the initial number of candidate solutions. This practice has been advocated in a number of competitions to compare the performance of population-based algorithms, and has been used in many articles that contain empirical comparisons of algorithms. This paper advocates the opinion that this practice does not result in fair comparisons, and provides an abundance of empirical evidence to support this claim. Empirical results are obtained from application of a standard global best particle swarm optimization (PSO) algorithm with different swarm sizes under the same FE computational limit, on a large benchmark suite.

Dynamic multi-objective optimisation problems have more than one objective with at least one objective that changes over time. Previous studies indicated that different knowledge sharing strategies increase the performance of the dynamic vector evaluated particle swarm optimisation (DVEPSO) algorithm in different dynamic environments. Therefore, this paper investigates the performance of the DVEPSO algorithm using heterogeneous particle swarm optimisation (HPSO) algorithms, where each particle uses a different knowledge sharing strategy. The goal of this study is to determine whether the use of HPSOs will improve the performance of DVEPSO by incorporating particles with different knowledge sharing strategies in a single DVEPSO algorithm. The results indicate that using HPSOs improves the performance of DVEPSO for dynamic multi-objective optimisation problems with a complex Pareto-optimal set and that the performance of heterogeneous DVEPSO compares favourably with that of DVEPSO.

Particle swarm optimisation (PSO) algorithms have been successfully used to solve many complex real-world optimisation problems. Since their introduction in 1995, the focus of research in PSOs has largely been on the algorithmic side with many new variations proposed on the original PSO algorithm. Relatively little attention has been paid to the study of problems with respect to PSO performance. The aim of this study is to investigate whether a link can be found between problem characteristics and algorithm performance for PSOs. A range of benchmark problems are numerically characterised using fitness landscape analysis techniques. Decision tree induction is used to develop failure prediction models for seven different variations on the PSO algorithm. Results show that for most PSO models, failure could be predicted to a fairly high level of accuracy. The resulting prediction models are not only useful as predictors of failure, but also provide insight into the algorithms themselves, especially when expressed as fuzzy rules in terms of fitness landscape features.

Applying weight regularisation to gradient-descent based neural network training methods such as backpropagation was shown to improve the generalisation performance of a neural network. However, the existing applications of weight regularisation to particle swarm optimisation are very limited, despite being promising. This paper proposes adding a regularisation penalty term to the objective function of the particle swarm. The impact of different penalty terms on the resulting neural network performance as trained by both backpropagation and particle swarm optimisation is analysed. Swarm behaviour under weight regularisation is studied, showing that weight regularisation results in smaller neural network architectures and more convergent swarms.

A new variant of the competitive coevolutionary team-based particle swarm optimiser (CCPSO(t)) algorithm is developed to train multi-agent teams from zero knowledge. Analysis show that the CCPSO algorithm stagnates during the training of simple soccer players. It is hypothesised that the stagnation is caused by saturation of the neural network weights. The CCPSO(t) algorithm is developed to overcome the stagnation problem. CCPSO(t) is based on the previously developed CCPSO algorithm with two additions. The first addition is the introduction of a restriction on the personal best particle positions. The second addition is the introduction of a linearly decreasing perception and core limit of the charged particle swarm optimiser. The final results show that the CCPSO(t) algorithm successfully addresses the CCPSO algorithm's neural network weight saturation problem.

Particle swarm optimization (PSO) algorithms have a number of parameters to which their behaviour is sensitive. In order to avoid problem-specific parameter tuning, a number of self-adaptive PSO algorithms have been proposed over the past few years. This paper compares the behaviour and performance of a selection of self-adaptive PSO algorithms to that of time-variant algorithms on a suite of 22 boundary constrained benchmark functions of varying complexities. It was found that only two of the nine selected self-adaptive PSO algorithms performed comparably to similar time-variant PSO algorithms. Possible reasons for the poor behaviour of the other algorithms as well as an analysis of the more successful algorithms is performed in this paper.

Many adaptations to the original particle swarm optimization algorithms have been developed to improve performance with respect to the quality of the solutions found, convergence speed, and robustness. One class of such adaptations incorporates evolutionary operators within the particle swarm optimization algorithm cycle. To date, selection, mutation, and crossover operators have been incorporated within particle swarm optimizers with varying degrees of success. This article focuses on particle swarm optimizers that utilize discrete crossover operators, with the main objective to show if any performance gains can be achieved by incorporating discrete crossover. Six discrete crossover operators are proposed for incorporation into a global best particle swarm optimizer. The performance of these discrete crossover operators are compared with that of the global best particle swarm optimizer and amongst one another to identify the best performing discrete crossover operators. The best operators are then compared with particle swarm optimizers that make use of blending crossover operators. Empirical evidence obtained from an extensive benchmark suite shows that two of the proposed discrete crossover operators perform significantly better than the global best particle swarm optimizer and all of the other crossover operators.

In temporal environments, population-based data clustering algorithms suffer when changes in the data occur during the clustering process. Diversity of the population is lost and memory of the individuals of the population is outdated, making the clusters found before the change non-optimal. This paper proposes a new particle swarm optimisation alternative to clustering temporal data. It combines the dynamic properties of the multi-swarm particle swarm optimisation algorithm with the multi-objective properties of the cooperative particle swarm optimisation algorithm. The proposed alternative is compared to various existing data clustering algorithms which are shortly described in the paper and the results are discussed, including a comparison of four performance measures relevant to the clustering of data.

This paper investigates the algorithm selection problem, otherwise referred to as the entity-to-algorithm allocation problem, within the context of three recent multi-method algorithm frameworks. A population-based algorithm portfolio, a meta-hyper-heuristic and a bandit based operator selection method are evaluated under similar conditions on a diverse set of floating-point benchmark problems. The meta-hyper heuristic is shown to outperform the other two algorithms.

Particle swarm optimization (PSO) is a well-known optimization technique originally proposed for solving single-objective, continuous optimization problems. However, PSO has been extended in various ways to handle multi-objective optimization problems (MOPs). The scalability of multi-objective PSO algorithms as the number of sub-objectives increases has not been well examined; most observations are for two to four objectives. It has been observed that the performance of multiobjective optimizers for a low number of sub-objectives can not be generalized to problems with higher numbers of sub-objectives. With this in mind, this paper presents a scalability study of three well-known multi-objective PSOs, namely vector evaluated PSO (VEPSO), optimized multi-objective PSO (oMOPSO), and speed-constrained multi-objective PSO (SMPSO) with up to eight sub-objectives. The study indicates that as the number of sub-objectives increases, SMPSO scaled the best, oMOPSO scaled the worst, while VEPSO's performance was dependent on the knowledge transfer strategy (KTS) employed, with parent centric recombination (PCX) based approaches scaling consistently better.

Dynamic multi-objective optimisation problems (DMOOPs) have more than one objective, with at least one objective changing over time. Since at least two of the objectives are normally in conflict with one another, a single solution does not exist and the goal of the algorithm is to track a set of tradeoff solutions over time. Analysing the performance of a dynamic multi-objective optimisation algorithm (DMOA) is not a trivial task. For each environment (before a change occurs) the DMOA has to find a set of solutions that are both diverse and as close as possible to the optimal trade-off solution set. In addition, the DMOA has to track the changing set of trade-off solutions over time. Approaches used to analyse the performance of dynamic single-objective optimisation algorithms (DSOAs) and DMOAs do not provide any information about the ability of the algorithms to track the changing optimum. Therefore, this paper introduces a new approach to analyse the performance of DMOAs and applies this approach to the results obtained by five DMOAs. In addition, it compares the new analysis approach to another approach that does not take the tracking ability of the DMOAs into account. The results indicate that the new analysis approach provide additional information, measuring the ability of the algorithm to find good performance measure values while tracking the changing optima.

This paper investigates whether the optimal parameter configurations for particle swarm optimizers (PSO) change when changes in the search landscape occur. To test this, specific environmental changes that may occur during dynamic function optimization are deliberately constructed, using the moving peaks function generator. The parameters of the chargedand quantum PSO algorithms are then optimized for the initial environment, as well as for each of the constructed problems. It is shown that the optimal parameter configurations for the various environments differ not only with respect to the initial optimal configurations, but also with respect to each other. The results lead to the conclusion that PSO parameters need to be re-optimized or selfadapted whenever environmental changes are detected.

Fitness landscape analysis has focussed on many different aspects of optimisation problems such as ruggedness, neutrality, epistasis and evolvability. Although many techniques have been proposed, there are very few that have been shown to be practically useful as predictors of algorithm performance. This paper investigates three metrics related to the structure of fitness landscapes of continuous problems: a ruggedness measure based on entropy, a dispersion index measure for detecting the presence of funnels and a new proposed technique for estimating gradients. Results on a range of benchmark problems show that all proposed measures show some correlation to performance of a traditional particle swarm optimisation (PSO) algorithm on the same benchmark problems. The three metrics could therefore have value as part-predictors of PSO performance on unknown problems if used in conjunction with measures approximating other features that have been linked to problem difficulty for PSOs.

Heterogeneous particle swarm optimizers (HPSO) allow particles to use different update equations, referred to as behaviors, within the swarm. Dynamic HPSOs allow the particles to change their behaviors during the search. These HPSOs alter the exploration/exploitation balance during the search which alters the search behavior of the swarm. This paper introduces a new self-adaptive HPSO and compares it with other HPSO algorithms on the CEC 2013 real-parameter optimization benchmark functions. The proposed algorithm keeps track of how successful each behavior has been over a number of iterations and uses that information to select the next behavior of a particle. The results show that the proposed algorithm outperforms existing HPSO algorithms on the benchmark functions.

The vector evaluated particle swarm optimization (VEPSO) algorithm is a cooperative, multi-swarm algorithm. Each sub-swarm optimizes only a single objective of a multi-objective problem (MOP), and implements a knowledge transfer strategy (KTS) to share optimal positions of the different objectives among the sub-swarms, guiding the particles to different regions of the Pareto front. This paper shows that the stagnation problem that occurs in VEPSO can be addressed by using a different KTS. A comparison is made between the ring-based and random knowledge transfer strategies. Experimental results show that the random knowledge transfer strategy suffers less from stagnation than the ring-based KTS, making it the preferred KTS to use.

Most optimization algorithms are designed to solve static, unchanging problems. However, many real-world problems exhibit dynamic behavior. Particle swarm optimization (PSO) is a successful metaheuristic methodology which has been adapted for locating and tracking optima in dynamic environments. Recently, a powerful new class of PSO strategies using cooperative principles was shown to improve PSO performance in static environments. While there exist many PSO algorithms designed for dynamic optimization problems, only one cooperative PSO strategy has been introduced for this purpose, and it has only been studied under one type of dynamism. This study proposes a new cooperative PSO strategy designed for dynamic environments. The newly proposed algorithm is shown to achieve significantly lower error rates when compared to well-known algorithms across problems with varying dimensionalities, temporal change severities, and spatial change severities.

Model aggregation is the process of constructing several base models that are then combined into a single model for prediction. Ensemble classification has been studied by many researchers and found to provide significant performance improvements over single models. This paper presents a new base model combination algorithm for K-nearest neighbor (KNN) ensemble models based on One-Versus-All (OVA) classification. The proposed algorithm uses two decision functions to determine the best prediction among the many predictions provided by the base models. It is demonstrated in this paper that tied or conflicting predictions can be effectively resolved when a probabilistic function and a distance function are used by a combination algorithm for OVA KNN base model predictions. The resolution of tied predictions leads to improvements in predictive performance.

The process of constructing several base models that are then combined into a single classification model for prediction is called model aggregation or ensemble classification. Positive-versus-negative (pVn) classification is a new method for the implementation of base models for aggregation. pVn classification involves the decomposition of a k-class prediction task into m (m < k) subproblems. One base model is constructed for each subproblem to predict a subset of the k classes. The base models are then combined into one aggregate model for prediction. This paper reports studies that were conducted to demonstrate the performance of pVn classification when large volumes of data are available for modeling as is commonly the case in data mining. It is demonstrated in this paper that pVn modeling provides the capability to use a large amount of available data (in a large data set) for base model training. It is also demonstrated that pVn models created from large data sets provide a higher level of predictive performance compared to single k-class models.

In spite of substantial research applying evolutionary algorithms and swarm based algorithms to solve dynamic problems, the classification of dynamic environments is missing universal standards. This paper examines the various methods used so far to characterise dynamic optimisation problems and proposes an inclusive classification system. Additionally, a way to generate environments of each type using the moving peak benchmark is described.

Since its birth in 1995, particle swarm optimization (PSO) has been well studied and successfully applied. While a better understanding of PSO and particle behaviors have been obtained through theoretical and empirical analysis, some issues about the beavior of particles remain unanswered. One such issue is how velocities should be initialized. Though zero initial velocities have been advocated, a popular initialization strategy is to set initial weights to random values within the domain of the optimization problem. This article first illustrates that particles tend to leave the boundaries of the search space irrespective of the initialization approach, resulting in wasted search effort. It is also shown that random initialization increases the number of roaming particles, and that this has a negative impact on convergence time. It is also shown that enforcing a boundary constraint on personal best positions does not help much to address this problem. The main objective of the article is to show that the best approach is to initialize particles to zero, or random values close to zero, without imposing a personal best bound.

This paper investigates the use of local search strategies to improve the performance of a meta-hyper-heuristic algorithm, a hyper-heuristic which employs one or more meta-heuristics as low-level heuristics. Alternative mechanisms for selecting the solutions to be refined further by means of local search, as well as the intensity of subsequent refinement in terms of number of allowable function evaluations, are investigated. Furthermore, defining a local search as one of the low-level heuristics versus applying the algorithm directly to the solution space is also investigated. Performance is evaluated on a diverse set of floating-point benchmark problems. The addition of local search was found to improve algorithm results significantly. Random selection of solutions for further refinement was identified as the best selection strategy and a higher intensity of refinement was identified as most desirable. Better results were obtained by applying the local search algorithm directly to the search space instead of defining it as a low-level heuristic.

The vector evaluated particle swarm optimisation (VEPSO) algorithm is a multi-swarm variation of particle swarm optimisation (PSO) used to solve static multi-objective optimisation problems (SMOOPs). Recently, VEPSO was extended to the dynamic VEPSO (DVEPSO) algorithm to solve dynamic multi-objective optimisation problems (DMOOPs) that have at least one objective that changes over time. The search process of DVEPSO is driven through local and global guides that can be updated in various ways. This paper investigates the influence of various guide update approaches on the performance of DVEPSO. DVEPSO is also compared against a competitive-cooperative evolutionary algorithm. The results indicate that DVEPSO performs well in fast changing environments, but struggles to converge to discontinuous Pareto-optimal fronts (POFs).

When solving a single objective optimization problem, a user desires an accurate solution, but may be computationally constrained in terms of the number of objective function evaluations (OFEs) that can be afforded. The OFE budget is application specific, varying depending on the time, computing resources, and the nature of the optimization problem. Control parameter value sensitivity to this OFE budget constraint is investigated for the particle swarm- and differential evolution optimization algorithms. The algorithms are tuned to selected testing problems under different OFE budget constraints, and then their performance is assessed at different OFE budgets from what they were tuned for. The results give evidence that combinations of optimization algorithm control parameter values which perform well for high OFE budgets do not perform well for low OFE budgets and vice versa. This indicates that when selecting control parameter values for these two algorithms, not only should the optimization problem characteristics be taken into account, but also the computational constraints.

Algorithm selection is an important consideration in multi-method global optimization. This paper investigates the use of various algorithm selection strategies derived from well known evolutionary selection mechanisms. Selection strategy performance is evaluated on a diverse set of floating point benchmark problems and meaningful conclusions are drawn with regard to the impact of selective pressure on algorithm selection in a multi-method environment.

Many optimisation problems have more than one objective that are in conflict with one another and that change over time, called dynamic multi-objective problems. To solve these problems an algorithm must be able to track the changing Pareto Optimal Front (POF) over time and find a diverse set of solutions. This requires detecting that a change has occurred in the environment and then responding to the change. Responding to the change also requires to update the archive of non-dominated solutions that represents the found POF. This paper discusses various ways to manage the archive solutions when a change occurs in the environment. Furthermore, two new benchmark functions are presented where the POF is discontinuous. The dynamic Vector Evaluation Particle Swarm Optimisation (DVEPSO) algorithm is tested against a variety of benchmark function types and its performance is compared against three state-of-the-art DMOO algorithms.

This paper investigates different trade decision strategies under different market conditions so that a genetic algorithm could be designed to use the appropriate decision strategy. A trade decision strategy defines how a single action is decided upon based on a number of signals where each signal is a result of a technical analysis function. Using historical market data, a population is trained using a simple genetic algorithm employing crossover and mutation. Four genetic algorithms are used to evolve agents to trade, where each genetic algorithm uses a different trade decision strategy. The best individual from each evolved population is compared using an out-of-sample data set. Results show a significant difference in performance between the four decision strategies especially within bearish to moderately bullish stocks. Populations evolved using a weighted decision strategy performs better than strategies that are not weighted when trading bearish to moderately bullish stocks. Non-weighted decision strategies appear to out-perform weighted strategies when used on extremely bullish stock. This out-performance could be attributed to fewer trades made by non-weighted strategies compared to weighted ones.

A competitive coevolutionary particle swarm optimization approach is proposed in this paper to train neural networks from zero knowledge to act as security trading agents. The coevolved neural networks are used for timing buying and short selling securities to maximize net profit and minimize risk over time. The proposed model attempts to identify security trend reversals using technical market indicators. No expert trading knowledge is presented to the model, only the technical market indicator data. A competitive fitness function is defined that allows the evaluation of each solution relative to other solutions, based on predefined performance metric objectives. The relative fitness function in this study considers net profit and the Sharpe ratio as a risk measure. For the purposes of this study, the stock prices of eight large market capitalisation companies were chosen. Two benchmarks were used to evaluate the discovered trading agents, consisting of a Bollinger Bands/Relative Strength Index rule-based strategy and the popular buy-and-hold strategy. The agents that were discovered from the proposed model outperformed both benchmarks by producing higher returns for in-sample and out-sample data at a low risk.

The purpose of this paper is to investigate the use of meta-heuristics as low-level heuristics in a hyper-heuristic framework. A novel multi-method hyper-heuristic algorithm which makes use of a number of common meta-heuristics is presented. Algorithm performance is evaluated on a diverse set of real parameter benchmark problems and meaningful conclusions are drawn with respect to the selection of alternative low-level heuristics and the acceptance of the obtained solutions within the proposed multi-method meta-heuristic approach.

This paper investigates the applicability of swarm-based algorithms to the game of Tetris. This work proposes an approach to the problem in which neural network weight values are optimized using a particle swarm optimization (PSO) algorithm. Such an approach has not previously been demonstrated as feasible for Tetris. The reported experimental results show the learning progress of the algorithm, as well as a comparison against a hand-optimized Tetris playing algorithm. The results indicate that the Tetris agents show a continuous improvement over the course of training. Since the experimental focus was on the feasibility of the approach rather than optimizing performance, optimized PSO-based agents were found to be outperformed by the hand-optimized algorithm. However, the playing strategies of the two agents were compared and shown to be similar. The results indicate that a swarm-based approach is feasible, and warrants further investigation.

The purpose of this paper is to investigate the overfitting behavior of particle swarm optimization (PSO) trained neural networks. Neural networks trained with PSOs using the global best, local best and Von Neumann information sharing topologies are investigated. Experiments are conducted on five classification and five time series regression problems. It is shown that differences exist in the degree of overfitting between the different topologies. Additionally, non-convergence of the swarms is witnessed, which is hypothetically attributed to the use of a bounded activation function in the neural networks. The hypothesis is supported by experiments conducted using an unbounded activation function in the neural network hidden layer, which lead to convergent swarms. Additionally this also lead to drastically reduced overfitting by the neural networks.

The Particle Swarm Optimiser (PSO) is a population based stochastic optimisation algorithm, empirically shown to be efficient and robust. This paper provides a proof to show that the original PSO does not have guaranteed convergence to a local optimum. A flaw in the original PSO is identified which causes stagnation of the swarm. Correction of this flaw results in a PSO algorithm with guaranteed convergence to a local minimum. Further extensions with provable global convergence are also described. Experimental results are provided to elucidate the behavior of the modified PSO as well as PSO variations with global convergence.

Recently a great emphasis has been given on autonomous decentralized networks (ADNs) wherein constituent nodes carry out specific tasks collectively. Their dynamic and constrained nature along with the emerging need for offering quality of service (QoS) assurances drive the necessity for effective network control mechanisms. This study focuses on designing a robust and self-adaptable congestion control mechanism which aims to be simple to implement at the individual node, and involve minimal information exchange, while maximizing network lifetime and providing QoS assurances. Our approach combats congestion by mimicking the collective behavior of bird flocks having global self-* properties achieved collectively without explicitly programming them into individual nodes. The main idea is to dasiaguidepsila packets (birds) to form flocks and flow towards the sink (global attractor), whilst trying to avoid congestion regions (obstacles). Unlike the bio-swarm approach of Couzin, which is formulated on a metrical space, our approach is reformulated on to a topological space (graph of nodes), while repulsion/attraction forces manipulate the direction of motion of packets. Our approach provides sink direction discovery, congestion detection and traffic management in ADNs with emphasis on Wireless Sensor Networks (WSNs). Performance evaluations show the effectiveness of our self-adaptable mechanism in balancing the offered load and in providing graceful performance degradation under high load scenarios compared to typical conventional approaches.

This paper introduces a new vector evaluated multi-objective optimization algorithm. The vector evaluated differential evolution particle swarm optimization (VEDEPSO) algorithm is a hybridization of the classical vector evaluated particle swarm optimization (VEPSO) and vector evaluated differential evolution (VEDE) algorithms of Parsopoulos et. al. Comparisons of VEDEPSO with respect to VEPSO and VEDE on a well known multi-objective benchmark problem set indicated that significant performance improvements can be attributed to the VEDEPSO algorithm.

A major unsolved problem in the field of optimisation and computational intelligence is how to determine which algorithms are best suited to solving which problems. This research aims to analytically characterise individual problems as a first step towards attempting to link problem types with the algorithms best suited to solving them. In particular, an information theoretic technique for analysing the ruggedness of a fitness landscape with respect to neutrality was adapted to work in continuous landscapes and to output a single measure of ruggedness. Experiments run on test functions with increasing ruggedness show that the proposed measure of ruggedness produced relative values consistent with a visual inspection of the problem landscapes. Combined with other measures of complexity, the proposed ruggedness measure could be used to more broadly characterise the complexity of fitness landscapes in continuous domains.

Most optimization algorithms suffer from a significant deterioration in performance as the dimensionality and complexity of the problem search space increases. Also these algorithms, given certain configurations, typically show markedly improved performance on a particular problem only to exhibit poor performance on another. The first issue could be resolved by using a cooperative algorithm to divide the problem complexity among its participating algorithms, making the problem easier to solve. The second issue could then be resolved with the use of differently configured participating algorithms within the overall cooperative algorithm. This paper investigates the possibility of combining different population-based algorithms within a cooperative algorithm. The aim is to take advantage of different algorithm characteristics regarding parameter settings, explorative/exploitative capacity, convergence speed and other behaviors in finding solutions to various optimization problems.

Free search differential evolution (FSDE) is a new, population-based meta-heuristic algorithm that is a hybrid of concepts from free search (FS), differential evolution (DE) and opposition-based learning. The performance of the proposed approach is investigated and compared with DE and one of the recent variants of DE when applied to ten benchmark functions. The experiments conducted show that FSDE provides excellent results with the added advantage of no parameter tuning.

Supervised neural networks (NNs) have been successfully applied to solve classification problems. Various NN training algorithms were developed, including the particle swarm optimiser (PSO), which was proved to outperform the standard back propagation training algorithm on a selection of problems. It was, however, usually assumed that the decision boundaries do not change over time. Such assumption is often not valid for real life problems, and training algorithms have to be adapted to track the changing decision boundaries and detect new boundaries as they appear. Various dynamic versions of the PSO have already been developed, and this paper investigates the applicability of dynamic PSO to NN training in changing environments.

This paper investigates the feasibility of using Genetic Programming in dynamically changing environments to evolve decision trees for classification problems and proposes an new version of Genetic Programming called Adaptive Genetic Programming. It does so by comparing the performance or classification error of Genetic Programming and Adaptive Genetic Programming to that of Gradient Descent in abruptly and progressively changing environments. To cope with dynamic environments, Adaptive Genetic Programming incorporates adaptive control parameters, variable elitism and culling. Results show that both Genetic Programming and Adaptive Genetic Programming are viable algorithms for dynamic environments yielding a performance gain over Gradient Descent for lower dimensional problems even with severe environment changes. In addition, Adaptive Genetic Programming performs slightly better than Genetic Programming, due to faster recovery from changes in the environment.

This paper presents an investigation into the scalability of the vector-based PSO, a niching algorithm using particle swarm optimization. The vector-based PSO locates and maintains niches by using vector operations to determine niche boundaries. The technique builds upon existing knowledge of the particle swarm in such a way that the swarm can be organized into subswarms without prior knowledge of the number of niches in the search space and the corresponding niche radii, thus reducing the number of user-specified parameters. In a designated search space a linear increase in the number of dimensions often results in an exponential or near exponential increase in the number of optima. Empirical results are reported where the vector-based PSO is tested on three multimodal functions in one to four dimensions using a range of swarm sizes. Optimal swarm sizes are derived where all or most of the optima should be located.

The self-organizing feature map (SOM) is an unsupervised neural network. It preserves a high-dimensional training data space's approximate characteristics, while scaling it to a two-dimensional grid. Few SOM-based rule extraction methods exist, and little analysis has been done on their overall viability. This paper presents the novel HybridSOMframework, which allows the combination of a SOM with any standard rule extraction algorithm, creating a customized hybrid rule extractor. Some HybridSOMvariations and traditional rule extraction algorithms are empirically compared, and the framework is critically discussed. This analysis also points to new conclusions on the viability of SOM-based rule extraction, in general.

This special issue contains seven papers describing recent research developments in the swarm intelligence (SI) field.

The theory of idiotopic lymphocyte networks in the natural immune system inspired the modelling of network based artificial immune systems (AIS). Many of these network based AIS models establish network links between the artificial lymphocytes (ALCs) whenever the measured Euclidean distance between the ALCs are below a certain network threshold. The linked ALCs represent an artificial lymphocyte network. Graaff and Engelbrecht introduced the Local Network Neighbourhood AIS (LNNAIS) [2]. The interpretation of the network theory is the main difference between LNNAIS and existing network based AIS models. The LNNAIS uses the concept of an artificial lymphocyte neighbourhood to determine network links between ALCs [A.J. Graaf and A.P. Engelbrecht, 2007]. The purpose of this paper is to highlight the drawbacks of the proposed LNNAIS model and to address these drawbacks with some enhancements, improving LNNAIS towards a self regulating AIS.

Many optimisation problems are multi-objective and change dynamically. Many methods use a weighted average approach to the multiple objectives. This paper introduces the usage of the vector evaluated particle swarm optimiser (VEPSO) to solve dynamic multi-objective optimisation problems. Every objective is solved by one swarm and the swarms share knowledge amongst each other about the objective that it is solving. Not much work has been done on using this approach in dynamic environments. This paper discusses this approach as well as the effect of the population size and the response methods to a detected change on the performance of the algorithm. The results showed that more non-dominated solutions, as well as more uniformly distributed solutions, are found when all swarms are re-intialised when a change is detected, instead of only the swarm(s) optimising the specific objective function(s) that has changed. Furthermore, an increase in population size results in a higher number of non-dominated solutions found, but can lead to solutions that are less uniformly distributed.

This paper investigates the application of alternative multi-objective optimization (MOO) strategies to a complex scheduling problem. Two vector evaluated algorithms, namely the vector evaluated particle swarm optimization (VEPSO) algorithm as well as the vector evaluated differential evolution (VEDE) algorithm is compared to a differential evolution based modified goal programming approach. This paper is considered significant since no other reference to the application of vector evaluated algorithms in a scheduling environment could be found. Algorithm performance is evaluated on real customer data and meaningful conclusions are drawn with respect to the application of MOO algorithms in a multiple machine multi-objective scheduling environment.

In studies that compare the performance of population-based optimization algorithms, it is sometimes assumed that the comparison is valid as long as the number of function evaluations is equal, even if the population size differs. This paper shows that such comparisons are invalid. The performance of two algorithms: differential evolution (DE) and global best particle swarm optimization (gbest PSO) are tested on standard benchmark problems with different numbers of individuals/particles (20, 50 and 100). It is shown that there are significance differences in the performance of the same algorithm with the same number of function evaluations, but with different numbers of individuals/particles. Comparisons of different algorithms should therefore always use the same population size for results to be valid.

An important factor contributing to the success of particle swarm optimization (PSO) is the balance between exploration and exploitation of the swarm. Exploration is typically preferred at the initial stages of the search but is required to gradually give way to exploitation of promising solutions as the search progresses. The diversity of a particle swarm optimization algorithm can be defined, simply, as the degree of dispersion of the particles in the swarm. This dispersion could be defined around some center-point or not. It could also be defined based on the positions of the particles or on their velocities. This paper takes a look at some of the different definitions of swarm diversity with the intention of determining their usefulness in quantifying swarm exploration/exploitation. This work is intended to lay the foundations for the development of a suitable means to quantify the rate of change from exploration to exploitation of a PSO, i.e. the rate of change of diversity.

This article reports improvements on DynDE, a approach to using Differential Evolution to solve dynamic optimization problems. Three improvements are suggested, namely favored populations, migrating individuals and a combination of these approaches. The effects of varying the change frequency, peak widths and the number of dimensions of the dynamic environment are investigated. Experimental results are presented that indicate that the suggested approaches constitute considerable improvements on previous research.

Most optimisation algorithms from the computational intelligence field assume that the search landscape is static. However, this assumption is not valid for many real-world problems. Therefore, there is a need for efficient optimisation algorithms that can track changing optima. A number of variants of particle swarm optimisation (PSO) have been developed for dynamic environments. Recently, the cooperative PSO has been shown to significantly improve performance of PSO in static environments, especially for high-dimensional problems. This paper investigates the performance of a cooperative version of the charged PSO on a benchmark of dynamic optimisation problems. Empirical results show that the cooperative charged PSO is an excellent alternative to track dynamically changing optima.

CIlib is a recently developed open source library of computational intelligence (CI) algorithms. Developed in Java, and designed to be a generic framework of pluggable components, CIlib provides the CI researcher with a powerful tool to facilitate research in new CI techniques, and to easily benchmark against existing CI algorithms on a variety of problems. Consisting of a number of frameworks, including a framework for most CI paradigms, CIlib also allows components from different frameworks to be weaved together to form hybrid CI models. This paper provides a detailed illustration of how CIlib can be used to easily setup simulations using different algorithms to solve various problems.

The self-organizing map (SOM) is an unsupervised neural network approach that reduces a high-dimensional data set to a representative and compact two-dimensional grid. In so doing, a SOM reveals emergent clusters within the data. Research has shown that SOMs lend themselves to visual and computational analysis for exploratory and data mining purposes. However, an important requirement for many SOM interpretations is the characterization of the mappsilas emergent clusters. This process is often addressed by either a manual or automated map neuron labeling approach. This paper discusses techniques for the labeling of the unsupervised, supervised and semi-supervised variants of the SOM, and proposes some new methods. It also presents empirical results characterizing the performance of two automated labeling approaches for fully unsupervised SOMs when applied for example classification of experimental data sets.

Research in computational intelligence (CI) has produced a huge collection of algorithms, grouped into the main CI paradigms. Development of a new CI algorithm requires such algorithm to be thoroughly benchmarked against existing algorithms, which requires researchers to implement already published algorithms. This re-implementation of existing algorithms unnecessarily wastes valuable time, and may be the cause of incorrect results due to unexpected bugs in the code. It is also the case that more, new CI algorithms are hybrids of algorithms from different paradigms. This illustrates a demand for a comprehensive library of CI algorithms, to minimize development time and the occurrence of programming errors, and to facilitate combination of components to form hybrid models. This paper presents such a library, called CIlib.

Ant colony optimization (ACO) is a powerful optimization technique that has been applied to solve a number of complex optimization problems. One such optimization problem is network topology design of distributed local area networks (DLANs). The problem requires simultaneous optimization of a number of objectives, such as monetary cost, average network delay, hop count between communicating nodes, and reliability under a set of constraints. This paper presents a multi-objective ant colony optimization algorithm to efficiently solve the DLAN topology design problem. The multi-objective aspect of the problem is handled by incorporating fuzzy logic in the ACO algorithm. The performance of fuzzy ACO is evaluated through comparison with a fuzzy simulated annealing algorithm. Empirical results suggest that the fuzzy ACO produces results of equal quality when compared with a fuzzy simulated annealing algorithm.

Neighbourhood topologies in particle swarm optimization (PSO) are typically random in terms of the spatial positions of connected neighbours. This study explores the use of spatially meaningful neighbours for PSO. An approach is designed which uses heuristics to leverage the natural neighbours computed with Delaunay triangulation. The approach is compared to standard PSO sociometries and fitness distance ratio approaches. Although intrinsic properties of Delaunay triangulation limit the practical application of this approach to low dimensions results show that it is a successful particle swarm optimizer.

The NichePSO was developed as one of the first particle swarm optimization (PSO) approaches to locate multiple solutions to continuous optimization problems. The NichePSO forms subswarms from a main swarm, where each subswarm represents a single niche (or solution). Mechanisms are employed to merge subswarms if they converge to the same solution, and also to absorb any particle within a subswarm if that particle enters the area covered by the subswarm. The NichePSO has shown very good performance in locating a good number of solutions to multimodal problems. However, it was found that the current subswarm merging and particle absorption strategies are premature, and limits exploration in the main swarm. This paper proposes a number of different merging and absorption strategies, and shows that fine tuning of these processes improves the performance of NichePSO on lower dimensional problems.

Differential evolution has shown to be a very powerful, yet simple, population-based optimization approach. The nature of its reproduction operator limits its application to continuous-valued search spaces. However, a simple discretization procedure can be used to convert floating-point solution vectors into discrete-valued vectors. This paper considers three approaches in which differential evolution can be used to solve problems with binary-valued parameters. The first approach is based on a homomorphous mapping, while the second approach interprets the floating-point solution vector as a vector of probabilities, used to decide on the appropriate binary value. The third approach normalizes solution vectors and then discretize these normalized vectors to form a bitstring. Empirical results are provided to illustrate the efficiency of both methods in comparison with particle swarm optimizers.

The artificial immune system (AIS) is inspired by the functioning of the natural immune system. There are different theories with regards to the organisational behaviour of the natural immune system. One of these theories is the network theory. In this paper a novel network based AIS model is proposed. The proposed Local Network Neighbourhood Artificial Immune System (LNNAIS) is inspired by the network topology of lymphocytes to learn the antigen structure from one another. LNNAIS has a different interpretation of the network theory compared to existing network based AIS models. LNNAIS uses a concept of anartificiallymphocyte(ALC)neighbourhoodto determine the network links between the ALCs. The purpose of this paper is to provide a proof of concept that anartificiallymphocyte(ALC)neighbourhoodcan cluster data in a dynamic environment. LNNAIS only requires one pass through the training data of antigen patterns for clustering.

A possible way of dealing with a high dimensional problem space is to divide it up into smaller parts, and to have each part optimized by a separate population. A mechanism is then defined to construct a complete solution from the subpopulations, and to evaluate the entities contained in the subpopulations. This form of cooperation has been successfully applied to particle swarm optimization (PSO), by [1] in the cooperative split PSO, and to genetic algorithms, in the cooperative coevolutionary genetic algorithm, developed by [2], on which the cooperative split PSO is based. This paper investigates cooperation in differential evolution (DE) with the aim of determining the effects of multiple participants in dealing with high-dimensional problem spaces.

The performance of two recent variants of differential evolution (DE) when applied to integer programming problems is investigated. The two DE variants, namely, self-adaptive DE (SDE) and DE using the ring neighborhood topology (a.k.a. DE/lbest/1) are compared with the standard DE and particle swarm optimization (PSO) methods on several integer programming test problems. The results show that the SDE seems to be an efficient alternative for solving integer programming problems.

Differential evolution (DE) is generally considered as a reliable, accurate, robust and fast optimization technique. DE has been successfully applied to solve a wide range of numerical optimization problems. However, the user is required to set the values of the control parameters of DE for each problem. Such parameter tuning is a time consuming task. In this paper, a new version of DE which eliminates the need for manual parameter tuning is proposed. The performance of the proposed approach is investigated and compared with other well-known approaches. The results show that the new algorithm provides good performance when applied to multimodal problems with the added advantage that no parameter tuning is needed.

A new, almost parameter-free optimization algorithm is developed in this paper as a hybrid of the barebones particle swarm optimizer (PSO) and differential evolution (DE). The DE is used to mutate, for each particle, the attractor associated with that particle, defined as a weighted average of its personal and neighborhood best positions. Results of this algorithm are compared to that of the barebones PSO, Von Neumann PSO, a DE PSO, and DE/rand/1/bin. These results show that the new algorithm provides excellent results with the added advantage that no parameter tuning is needed

The performance of two variants of particle swarm optimization (PSO) when applied to integer programming problems is investigated. The two PSO variants, namely, barebones particle swarm (BB) and the exploiting barebones particle swarm (BBExp) are compared with the standard PSO and standard differential evolution (DE) on several integer programming test problems. The results show that the BBExp seems to be an efficient alternative for solving integer programming problems

Reducing the dimensionality of high-dimensional data simplifies how data is presented, allowing easier visualisation of high-dimensional data and facilitating more efficient extraction of knowledge. Nonlinear mapping methods transform data existing in high-dimensional space into a lower-dimensional space such that the topological characteristics of the high-dimensional data are preserved. Recent work proposed a particle swarm optimisation algorithm to perform nonlinear mapping. This paper compares a number of optimisation algorithms in performing nonlinear mapping. Experimental results distinguish between each of the optimisation algorithms. Nonlinear mapping methods were designed to map small datasets and are unable to project new data points. A proposed method to perform nonlinear mapping on large datasets is discussed.

Determining RNA secondary structure computationally, rather than manually, has the advantage of being cheaper and quicker. This paper introduces a new set-based particle swarm optimization algorithm to optimize the structure of an RNA molecule, using an advanced thermodynamic model. Results show that it is possible to use this SetPSO algorithm to optimise RNA secondary structure and produce candidate RNA conformations.

The ability of differential evolution (DE) to perform well in continuous-valued search spaces is well documented. The arithmetic reproduction operator used by differential evolution is simple, however, the manner in which the operator is defined, makes it practically impossible to effectively apply the standard DE to other problem spaces. An interesting and unique mapping method is examined which will enable the DE algorithm to operate within binary space. Using angle modulation, a bit string can be generated using a trigonometric generating function. The DE is used to evolve the coefficients to the trigonometric function, thereby allowing a mapping from continuous-space to binary-space. Instead of evolving the higher-dimensional binary solution directly, angle modulation is used together with DE to reduce the complexity of the problem into a 4-dimensional continuous-valued problem. Experimental results indicate the effectiveness of the technique and the viability for the DE to operate in binary space.

Large datasets consisting of high-dimensional vectors commonly describe complex objects. Having these vectors exist in a smaller dimension where the topological characteristics of the original space are preserved, allows clusters or patterns inherent in the data to be identified. This paper investigates the capability of various particle swarm optimisation (PSO) structures to effectively map a high-dimensional dataset to a lower-dimensional set. Four different local nonlinear mapping methods are investigated. Results obtained from the experiments give a clear indication of which nonlinear method to use when certain conditions hold

The underlying relationship between various PSO parameters is experimentally examined by applying the binary PSO (BinPSO) algorithm to solve the knights cover problem. An exhaustive analysis of the cognitive and social acceleration constants is performed, as well as an investigation into the influence of an increased maximum velocity on overall performance. An intuitive visualisation method eases the analysis of experimental results, and certain assumptions about the direct mapping of continuous PSO to BinPSO parameter values are corrected. The effects of increasing the complexity of the problem are also directly studied and recommendations made to improve performance under larger board sizes

A clustering method that is based on differential evolution is developed in this paper. The algorithm finds the centroids of a user specified number of clusters, where each cluster groups together similar patterns. The application of the proposed clustering algorithm to the problem of unsupervised classification and segmentation of images is investigated. To illustrate its wide applicability, the proposed algorithm is then applied to synthetic, MRI and satellite images. Experimental results show that the differential evolution clustering algorithm performs very well compared to other state-of-the-art clustering algorithms in all measured criteria. Additionally, the paper presents a different formulation to the multi-objective fitness function to eliminate the need to tune objective weights. A gbest DE is also proposed with encouraging results.

The optimisation process of a particular problem generally has many influencing factors including the parameter choices, problem constraints as well as the complexity of the optimisation algorithm and optimisation problem among others. The dimensionality of a problem influences the computational complexity in converging to a valid solution. With problems defined in larger and more abstract dimensions, complexity becomes a problem as the solutions presented by the algorithm are more likely to be sub-optimal. An interesting and unique manner to reduce the complexity of binary problems is developed in this paper: angle modulation is applied to generate a bit string to solve binary problems, using particle swarm optimisation (PSO) to evolve the function coefficients of a trigonometric model. Instead of evolving a high dimensional bit vector, angle modulation reduces the problem to a four-dimensional problem defined in continuous space. Experimental results show that the angle modulation method is faster than the standard binary PSO, and that accuracy is improved for most benchmark functions used

Niching algorithms have the ability to locate and maintain more than one solution to a multi-modal optimization problem. Recently, niching algorithms have been developed for particle swarm optimization (PSO) to locate multiple optima. This paper investigates the ability of the basic PSO to locate and maintain niches, in order to arrive at a conclusion on whether special purpose PSO algorithms, like NichePSO, need to be developed at all. The main finding is that, due to the social component of the velocity update, the gbest PSO is incapable of niching, while the lbest PSO is inefficient in this task.

This paper presents CiClops, which is a virtual laboratory for performing experiments, using computational intelligence (CI) algorithms that scale over multiple workstations. Additionally, the paper introduces CIlib, which is an open source library of CI algorithms, currently containing mostly particle swarm optimization (PSO) and ant colony optimization (ACO) algorithms. The main purpose of CiClops is to specify CI algorithms to solve optimization problems, to schedule execution of large numbers of simulations on a cluster of workstations, and to archive all empirical data for analysis. The objective of this paper is to launch both CiClops and CIlib, and to emphasize to the CI (most specifically the swarm intelligence) research community the advantages of using these tools.

An image clustering method that is based on the particle swarm optimizer (PSO) is developed in this paper. The algorithm finds the centroids of a user specified number of clusters, where each cluster groups together with similar image primitives. To illustrate its wide applicability, the proposed image classifier has been applied to synthetic, MRI and satellite images. Experimental results show that the PSO image classifier performs better than state-of-the-art image classifiers (namely, K-means, Fuzzy C-means, K-Harmonic means and Genetic Algorithms) in all measured criteria. The influence of different values of PSO control parameters on performance is also illustrated.

This paper presents and investigates the application of coevolutionary training techniques based on particle swarm optimization (PSO) to evolve playing strategies for the nonzero sum problem of the iterated prisoner's dilemma (IPD). Three different coevolutionary PSO techniques are used, differing in the way that IPD strategies are presented: A neural network (NN) approach in which the NN is used to predict the next action, a binary PSO approach in which the particle represents a complete playing strategy, and finally, a novel approach that exploits the symmetrical structure of man-made strategies. The last technique uses a PSO algorithm as a function approximator to evolve a function that characterizes the dynamics of the IPD. These different PSO approaches are compared experimentally with one another, and with popular man-made strategies. The performance of these approaches is evaluated in both clean and noisy environments. Results indicate that NNs cooperate well, but may develop weak strategies that can cause catastrophic collapses. The binary PSO technique does not have the same deficiency, instead resulting in an overall state of equilibrium in which some strategies are allowed to exploit the population, but never dominate. The symmetry approach is not as successful as the binary PSO approach in maintaining cooperation in both noisy and noiseless environments-exhibiting selfish behavior against the benchmark strategies and depriving them of receiving almost any payoff. Overall, the PSO techniques are successful at generating a variety of strategies for use in the IPD, duplicating and improving on existing evolutionary IPD population observations.

This paper investigates two different approaches using particle swarm optimisation (PSO) to evolve strategies for iterated prisoner's dilemma (IPD). Strategies evolved by the lesser known binary PSO algorithm are compared to strategies evolved by neural networks that were trained using PSO. Evolved strategies are compared against well-known game theory strategies, with positive results. The presence of noise during IPD interactions are also investigated, and evolved strategies are compared against the same well-known game theory strategies in a noisy environment.

A new competitive approach is developed for learning agents to play two-agent games. This approach uses particle swarm optimizers (PSO) to train neural networks to predict the desirability of states in the leaf nodes of a game tree. The new approach is applied to the TicTacToe game, and compared with the performance of an evolutionary approach. A performance criterion is defined to quantify performance against that of players making random moves. The results show that the new PSO-based approach performs well as compared with the evolutionary approach.

The particle swarm optimizer (PSO) is a stochastic, population-based optimization technique that can be applied to a wide range of problems, including neural network training. This paper presents a variation on the traditional PSO algorithm, called the cooperative particle swarm optimizer, or CPSO, employing cooperative behavior to significantly improve the performance of the original algorithm. This is achieved by using multiple swarms to optimize different components of the solution vector cooperatively. Application of the new PSO algorithm on several benchmark optimization problems shows a marked improvement in performance over the traditional PSO.

This paper investigates the effectiveness of various particle swarm optimiser structures to learn how to play the game of checkers. Co-evolutionary techniques are used to train the game playing agents. Performance is compared against a player making moves at random. Initial experimental results indicate definite advantages in using certain information sharing structures and swarm size configurations to successfully learn the game of checkers.

This paper proposes two new approaches to using PSO to cluster data. It is shown how PSO can be used to find the centroids of a user specified number of clusters. The algorithm is then extended to use K-means clustering to seed the initial swarm. This second algorithm basically uses PSO to refine the clusters formed by K-means. The new PSO algorithms are evaluated on six data sets, and compared to the performance of K-means clustering. Results show that both PSO clustering techniques have much potential.

A new PSO algorithm, the linear PSO (LPSO), is developed to optimise functions constrained by linear constraints of the form Ax = b. A crucial property of the LPSO is that the possible movement of particles through vector spaces is guaranteed by the velocity and position update equations. This property makes the LPSO ideal in optimising linearly constrained problems. The LPSO is extended to the converging linear PSO, which is guaranteed to always find at least a local minimum.

This article is a proposal, by the Computational Intelligence Research Group at the University of Pretoria (CIRG@UP), for a framework to benchmark optimisation algorithms. This framework, known as OptiBench, was conceived out of the necessity to consolidate the efforts of a large research group. Many problems arise when different people work independently on their own research initiatives. These problems range from duplicating effort to, more seriously, having conflicting results. In addition, less experienced members of the group are sometimes unfamiliar with the necessary statistical methods required to properly analyse their results. These problems are not limited internally to CIRG@UP but are also prevalent in the research community at large. This proposal aims to standardise the research methodology used by CIRG@UP internally (initially in the optimisation subgroup and later in subgroups working in other paradigms of computational research). Obviously this article cannot dictate the methodologies that should be used by other members of the broader research community, however, the hope is that this framework can be found useful and that others would willingly contribute and become involved.

Training a support vector machine requires solving a constrained quadratic programming problem. Linear particle swarm optimization is intuitive and simple to implement, and is presented as an alternative to current numeric SVM training methods. Performance of the new algorithm is demonstrated on the MNIST character recognition dataset.

In contrast to optimization techniques intended to find a single, global solution in a problem domain, niching (speciation) techniques have the ability to locate multiple solutions in multimodal domains. Numerous niching techniques have been proposed, broadly classified as temporal (locating solutions sequentially) and parallel (multiple solutions are found concurrently) techniques. Most research efforts to date have considered niching solutions through the eyes of genetic algorithms (GA), studying simple multimodal problems. Little attention has been given to the possibilities associated with emergent swarm intelligence techniques. Particle swarm optimization (PSO) utilizes properties of swarm behaviour not present in evolutionary algorithms such as GA, to rapidly solve optimization problems. This paper investigates the ability of two genetic algorithm niching techniques, sequential niching and deterministic crowding, to scale to higher dimensional domains with large numbers of solutions, and compare their performance to a PSO-based niching technique, Niche PSO.

The standard particle swarm optimiser (PSO) may prematurely converge on suboptimal solutions that are not even guaranteed to be local extrema. The guaranteed convergence modifications to the PSO algorithm ensure that the PSO at least converges on a local extremum at the expense of even faster convergence. This faster convergence means that less of the search space is explored reducing the opportunity of the swarm to find better local extrema. Various neighbourhood topologies inhibit premature convergence by preserving swarm diversity during the search. This paper investigates the performance of the guaranteed convergence PSO (GCPSO) using different neighbourhood topologies and compares the results with their standard PSO counterparts.

Architecture selection is a very important aspect in the design of neural networks (NNs) to optimally tune performance and computational complexity. Sensitivity analysis has been used successfully to prune irrelevant parameters from feedforward NNs. This paper presents a new pruning algorithm that uses the sensitivity analysis to quantify the relevance of input and hidden units. A new statistical pruning heuristic is proposed, based on the variance analysis, to decide which units to prune. The basic idea is that a parameter with a variance in sensitivity not significantly different from zero, is irrelevant and can be removed. Experimental results show that the new pruning algorithm correctly prunes irrelevant input and hidden units. The new pruning algorithm is also compared with standard pruning algorithms.

A new evolutionary search algorithm, called BGP (Building-block approach to Genetic Programming), to be used for classification tasks in data mining, is introduced. It is different from existing evolutionary techniques in that it does not use indirect representations of a solution, such as bit strings or grammars. The algorithm uses decision trees of various sizes as individuals in the populations and operators, e.g. crossover, are performed directly on the trees. When compared to the C4.5 and CN2 induction algorithms on a benchmark set of problems, BGP shows very good results.

Product units in the hidden layer of multilayer neural networks provide a powerful mechanism for neural networks to efficiently learn higher-order combinations of inputs. Training product unit networks using local optimization algorithms is difficult due to an increased number of local minima and increased chances of network paralysis. The paper discusses the problems with using gradient descent to train product unit neural networks, and shows that particle swarm optimization, genetic algorithms and LeapFrog are efficient alternatives to successfully train product unit neural networks.

A new learning algorithm is presented that learns a function and its first-order derivatives. Derivatives are learned together with the function using gradient descent. Preliminary results show that the algorithm accurately approximates the derivatives.

A new incremental learning algorithm for function approximation problems is presented where the neural network learner dynamically selects during training the most informative patterns from a candidate training set. The incremental learning algorithm uses its current knowledge about the function to be approximated, in the form of output sensitivity information, to incrementally grow the training set with patterns that have the highest influence on the learning objective.

This paper presents an algorithm for pruning feedforward neural network architectures using sensitivity analysis. Sensitivity Analysis is used to quantify the relevance of input and hidden units. A new statistical pruning heuristic is proposed, based on the variance analysis, to decide which units to prune. Results are presented to show that the pruning algorithm correctly prunes irrelevant input and hidden units.