Segment-Based Predominant Learning Swarm Optimizer for Large-Scale Optimization

Neural Net-Enhanced Competitive Swarm Optimizer for Large-Scale Multiobjective Optimization

The competitive swarm optimizer (CSO) classifies swarm particles into loser and winner particles and then uses the winner particles to efficiently guide the search of the loser particles. This approach has very promising performance in solving large-scale multiobjective optimization problems (LMOPs). However, most studies of CSOs ignore the evolution of the winner particles, although their quality is very important for the final optimization performance. Aiming to fill this research gap, this article proposes a new neural net-enhanced CSO for solving LMOPs, called NN-CSO, which not only guides the loser particles via the original CSO strategy, but also applies our trained neural network (NN) model to evolve winner particles. First, the swarm particles are classified into winner and loser particles by the pairwise competition. Then, the loser particles and winner particles are, respectively, treated as the input and desired output to train the NN model, which tries to learn promising evolutionary dynamics by driving the loser particles toward the winners. Finally, when model training is complete, the winner particles are evolved by the well-trained NN model, while the loser particles are still guided by the winner particles to maintain the search pattern of CSOs. To evaluate the performance of our designed NN-CSO, several LMOPs with up to ten objectives and 1000 decision variables are adopted, and the experimental results show that our designed NN model can significantly improve the performance of CSOs and shows some advantages over several state-of-the-art large-scale multiobjective evolutionary algorithms as well as over model-based evolutionary algorithms.

A Two-Stage Swarm Optimizer With Local Search for Water Distribution Network Optimization

Evolutionary computation (EC) algorithms have been successfully applied to the small-scale water distribution network (WDN) optimization problem. However, due to the city expansion, the network scale grows at a fast speed so that the efficacy of many current EC algorithms degrades rapidly. To solve the large-scale WDN optimization problem effectively, a two-stage swarm optimizer with local search (TSOL) is proposed in this article. To address the issues caused by the large-scale and multimodal characteristics of the problem, the proposed algorithm divides the optimization process into an exploration stage and an exploitation stage. It first finds a promising region of the search space in the exploration stage. Then, it searches thoroughly in the promising region to obtain the final solution in the exploitation stage. To search effectively the huge search space, we propose an improved level-based learning optimizer and use it in both the exploration and exploitation stages. Two new local search algorithms are proposed to further improve the quality of the solution. Experiments on both synthetic benchmark networks and a real-world network show that the proposed algorithm has outperformed the state-of-the-art metaheuristic algorithms.

The Improved Cooperative Particle Swarm Optimization (ICPSO) with Dynamic Information Adjustment and Controllable Speed and Its Application in Neural Network Optimization

In view of the premature convergence of particle swarm optimization (PSO) that is often caused by the loss of diversity, an improved cooperative PSO (ICPSO) is proposed. The method can dynamically combine the optimum values of the particles themselves, the global particles and the optimum values in groups, use the current optimization stage to dynamically adjust the shared proportion of information and effectively fuse various reference information, which can obtain superior global and local optimization performance. Additionally, to improve the diversity of the algorithm, a dynamic adjustment method using the grouping coefficient [Formula: see text] for the convergence rate is put forward. This method makes the algorithm have a more appropriate convergence rate while improving the convergence precision and enhancing the performance of the algorithm. Finally, the algorithm is used to optimize a neural network. The convergence condition and convergence rate of the algorithm are assessed by theoretical analysis and simulation experiments. The results show that ICPSO has more advantages in its diversity and the adjustment of the convergence rate compared to other related algorithms. Regarding neural network optimization, the training speed and optimization precision of the ICPSO-BP neural network are the highest, which has reached the best and average level of classification accuracy 98.5%, 96.3% for 20 iterations in Iris, and 98.7%, 95.1% in Wine. Its average iteration times score the best in five problems out of six.

Predominant Cognitive Learning Particle Swarm Optimization for Global Numerical Optimization

Particle swarm optimization (PSO) has witnessed giant success in problem optimization. Nevertheless, its optimization performance seriously degrades when coping with optimization problems with a lot of local optima. To alleviate this issue, this paper designs a predominant cognitive learning particle swarm optimization (PCLPSO) method to effectively tackle complicated optimization problems. Specifically, for each particle, a new promising exemplar is constructed by letting its personal best position cognitively learn from a better personal experience randomly selected from those of others based on a novel predominant cognitive learning strategy. As a result, different particles preserve different guiding exemplars. In this way, the learning effectiveness and the learning diversity of particles are expectedly improved. To eliminate the dilemma that PCLPSO is sensitive to the involved parameters, we propose dynamic adjustment strategies, so that different particles preserve different parameter settings, which is further beneficial to promote the learning diversity of particles. With the above techniques, the proposed PCLPSO could expectedly compromise the search intensification and diversification in a good way to search the complex solution space properly to achieve satisfactory performance. Comprehensive experiments are conducted on the commonly adopted CEC 2017 benchmark function set to testify the effectiveness of the devised PCLPSO. Experimental results show that PCLPSO obtains considerably competitive or even much more promising performance than several representative and state-of-the-art peer methods.

Elite Directed Particle Swarm Optimization with Historical Information for High-Dimensional Problems

High-dimensional optimization problems are ubiquitous in every field nowadays, which seriously challenge the optimization ability of existing optimizers. To solve this kind of optimization problems effectively, this paper proposes an elite-directed particle swarm optimization (EDPSO) with historical information to explore and exploit the high-dimensional solution space efficiently. Specifically, in EDPSO, the swarm is first separated into two exclusive sets based on the Pareto principle (80-20 rule), namely the elite set containing the top best 20% of particles and the non-elite set consisting of the remaining 80% of particles. Then, the non-elite set is further separated into two layers with the same size from the best to the worst. As a result, the swarm is divided into three layers. Subsequently, particles in the third layer learn from those in the first two layers, while particles in the second layer learn from those in the first layer, on the condition that particles in the first layer remain unchanged. In this way, the learning effectiveness and the learning diversity of particles could be largely promoted. To further enhance the learning diversity of particles, we maintain an additional archive to store obsolete elites, and use the predominant elites in the archive along with particles in the first two layers to direct the update of particles in the third layer. With these two mechanisms, the proposed EDPSO is expected to compromise search intensification and diversification well at the swarm level and the particle level, to explore and exploit the solution space. Extensive experiments are conducted on the widely used CEC’2010 and CEC’2013 high-dimensional benchmark problem sets to validate the effectiveness of the proposed EDPSO. Compared with several state-of-the-art large-scale algorithms, EDPSO is demonstrated to achieve highly competitive or even much better performance in tackling high-dimensional problems.

Differential Elite Learning Particle Swarm Optimization for Global Numerical Optimization

Although particle swarm optimization (PSO) has been successfully applied to solve optimization problems, its optimization performance still encounters challenges when dealing with complicated optimization problems, especially those with many interacting variables and many wide and flat local basins. To alleviate this issue, this paper proposes a differential elite learning particle swarm optimization (DELPSO) by differentiating the two guiding exemplars as much as possible to direct the update of each particle. Specifically, in this optimizer, particles in the current swarm are divided into two groups, namely the elite group and non-elite group, based on their fitness. Then, particles in the non-elite group are updated by learning from those in the elite group, while particles in the elite group are not updated and directly enter the next generation. To comprise fast convergence and high diversity at the particle level, we let each particle in the non-elite group learn from two differential elites in the elite group. In this way, the learning effectiveness and the learning diversity of particles is expectedly improved to a large extent. To alleviate the sensitivity of the proposed DELPSO to the newly introduced parameters, dynamic adjustment strategies for parameters were further designed. With the above two main components, the proposed DELPSO is expected to compromise the search intensification and diversification well to explore and exploit the solution space properly to obtain promising performance. Extensive experiments conducted on the widely used CEC 2017 benchmark set with three different dimension sizes demonstrated that the proposed DELPSO achieves highly competitive or even much better performance than state-of-the-art PSO variants.

A Dimension Group-Based Comprehensive Elite Learning Swarm Optimizer for Large-Scale Optimization

High-dimensional optimization problems are more and more common in the era of big data and the Internet of things (IoT), which seriously challenge the optimization performance of existing optimizers. To solve these kinds of problems effectively, this paper devises a dimension group-based comprehensive elite learning swarm optimizer (DGCELSO) by integrating valuable evolutionary information in different elite particles in the swarm to guide the updating of inferior ones. Specifically, the swarm is first separated into two exclusive sets, namely the elite set (ES) containing the top best individuals, and the non-elite set (NES), consisting of the remaining individuals. Then, the dimensions of each particle in NES are randomly divided into several groups with equal sizes. Subsequently, each dimension group of each non-elite particle is guided by two different elites randomly selected from ES. In this way, each non-elite particle in NES is comprehensively guided by multiple elite particles in ES. Therefore, not only could high diversity be maintained, but fast convergence is also likely guaranteed. To alleviate the sensitivity of DGCELSO to the associated parameters, we further devise dynamic adjustment strategies to change the parameter settings during the evolution. With the above mechanisms, DGCELSO is expected to explore and exploit the solution space properly to find the optimum solutions for optimization problems. Extensive experiments conducted on two commonly used large-scale benchmark problem sets demonstrate that DGCELSO achieves highly competitive or even much better performance than several state-of-the-art large-scale optimizers.

Stochastic Triad Topology Based Particle Swarm Optimization for Global Numerical Optimization

Particle swarm optimization (PSO) has exhibited well-known feasibility in problem optimization. However, its optimization performance still encounters challenges when confronted with complicated optimization problems with many local areas. In PSO, the interaction among particles and utilization of the communication information play crucial roles in improving the learning effectiveness and learning diversity of particles. To promote the communication effectiveness among particles, this paper proposes a stochastic triad topology to allow each particle to communicate with two random ones in the swarm via their personal best positions. Then, unlike existing studies that employ the personal best positions of the updated particle and the neighboring best position of the topology to direct its update, this paper adopts the best one and the mean position of the three personal best positions in the associated triad topology as the two guiding exemplars to direct the update of each particle. To further promote the interaction diversity among particles, an archive is maintained to store the obsolete personal best positions of particles and is then used to interact with particles in the triad topology. To enhance the chance of escaping from local regions, a random restart strategy is probabilistically triggered to introduce initialized solutions to the archive. To alleviate sensitivity to parameters, dynamic adjustment strategies are designed to dynamically adjust the associated parameter settings during the evolution. Integrating the above mechanism, a stochastic triad topology-based PSO (STTPSO) is developed to effectively search complex solution space. With the above techniques, the learning diversity and learning effectiveness of particles are largely promoted and thus the developed STTPSO is expected to explore and exploit the solution space appropriately to find high-quality solutions. Extensive experiments conducted on the commonly used CEC 2017 benchmark problem set with different dimension sizes substantiate that the proposed STTPSO achieves highly competitive or even much better performance than state-of-the-art and representative PSO variants.

An Adaptive Stochastic Dominant Learning Swarm Optimizer for High-Dimensional Optimization

High-dimensional problems are ubiquitous in many fields, yet still remain challenging to be solved. To tackle such problems with high effectiveness and efficiency, this article proposes a simple yet efficient stochastic dominant learning swarm optimizer. Particularly, this optimizer not only compromises swarm diversity and convergence speed properly, but also consumes as little computing time and space as possible to locate the optima. In this optimizer, a particle is updated only when its two exemplars randomly selected from the current swarm are its dominators. In this way, each particle has an implicit probability to directly enter the next generation, making it possible to maintain high swarm diversity. Since each updated particle only learns from its dominators, good convergence is likely to be achieved. To alleviate the sensitivity of this optimizer to newly introduced parameters, an adaptive parameter adjustment strategy is further designed based on the evolutionary information of particles at the individual level. Finally, extensive experiments on two high dimensional benchmark sets substantiate that the devised optimizer achieves competitive or even better performance in terms of solution quality, convergence speed, scalability, and computational cost, compared to several state-of-the-art methods. In particular, experimental results show that the proposed optimizer performs excellently on partially separable problems, especially partially separable multimodal problems, which are very common in real-world applications. In addition, the application to feature selection problems further demonstrates the effectiveness of this optimizer in tackling real-world problems.

Stochastic Cognitive Dominance Leading Particle Swarm Optimization for Multimodal Problems

Optimization problems become increasingly complicated in the era of big data and Internet of Things, which significantly challenges the effectiveness and efficiency of existing optimization methods. To effectively solve this kind of problems, this paper puts forward a stochastic cognitive dominance leading particle swarm optimization algorithm (SCDLPSO). Specifically, for each particle, two personal cognitive best positions are first randomly selected from those of all particles. Then, only when the cognitive best position of the particle is dominated by at least one of the two selected ones, this particle is updated by cognitively learning from the better personal positions; otherwise, this particle is not updated and directly enters the next generation. With this stochastic cognitive dominance leading mechanism, it is expected that the learning diversity and the learning efficiency of particles in the proposed optimizer could be promoted, and thus the optimizer is expected to explore and exploit the solution space properly. At last, extensive experiments are conducted on a widely acknowledged benchmark problem set with different dimension sizes to evaluate the effectiveness of the proposed SCDLPSO. Experimental results demonstrate that the devised optimizer achieves highly competitive or even much better performance than several state-of-the-art PSO variants.

Subpopulation Particle Swarm Optimization with a Hybrid Mutation Strategy

With the large-scale optimization problems in the real world becoming more and more complex, they also require different optimization algorithms to keep pace with the times. Particle swarm optimization algorithm is a good tool that has been proved to deal with various optimization problems. Conventional particle swarm optimization algorithms learn from two particles, namely, the best position of the current particle and the best position of all particles. This particle swarm optimization algorithm is simple to implement, simple, and easy to understand, but it has a fatal defect. It is hard to find the global optimal solution quickly and accurately. In order to deal with these defects of standard particle swarm optimization, this paper proposes a particle swarm optimization algorithm (SHMPSO) based on the hybrid strategy of seed swarm optimization (using codes available from https://gitee.com/mr-xie123234/code/tree/master/). In SHMPSO, a subpopulation coevolution particle swarm optimization algorithm is adopted. In SHMPSO, an elastic candidate-based strategy is used to find a candidate and realize information sharing and coevolution among populations. The mean dimension learning strategy can be used to make the population converge faster and improve the solution accuracy of SHMPSO. Twenty-one benchmark functions and six industries-recognized particle swarm optimization variants are used to verify the advantages of SHMPSO. The experimental results show that SHMPSO has good convergence speed and good robustness and can obtain high-precision solutions.

An Adaptive Covariance Scaling Estimation of Distribution Algorithm

Optimization problems are ubiquitous in every field, and they are becoming more and more complex, which greatly challenges the effectiveness of existing optimization methods. To solve the increasingly complicated optimization problems with high effectiveness, this paper proposes an adaptive covariance scaling estimation of distribution algorithm (ACSEDA) based on the Gaussian distribution model. Unlike traditional EDAs, which estimate the covariance and the mean vector, based on the same selected promising individuals, ACSEDA calculates the covariance according to an enlarged number of promising individuals (compared with those for the mean vector). To alleviate the sensitivity of the parameters in promising individual selections, this paper further devises an adaptive promising individual selection strategy for the estimation of the mean vector and an adaptive covariance scaling strategy for the covariance estimation. These two adaptive strategies dynamically adjust the associated numbers of promising individuals as the evolution continues. In addition, we further devise a cross-generation individual selection strategy for the parent population, used to estimate the probability distribution by combing the sampled offspring in the last generation and the one in the current generation. With the above mechanisms, ACSEDA is expected to compromise intensification and diversification of the search process to explore and exploit the solution space and thus could achieve promising performance. To verify the effectiveness of ACSEDA, extensive experiments are conducted on 30 widely used benchmark optimization problems with different dimension sizes. Experimental results demonstrate that the proposed ACSEDA presents significant superiority to several state-of-the-art EDA variants, and it preserves good scalability in solving optimization problems.

A Two-Phase Learning-Based Swarm Optimizer for Large-Scale Optimization

In this article, a simple yet effective method, called a two-phase learning-based swarm optimizer (TPLSO), is proposed for large-scale optimization. Inspired by the cooperative learning behavior in human society, mass learning and elite learning are involved in TPLSO. In the mass learning phase, TPLSO randomly selects three particles to form a study group and then adopts a competitive mechanism to update the members of the study group. Then, we sort all of the particles in the swarm and pick out the elite particles that have better fitness values. In the elite learning phase, the elite particles learn from each other to further search for more promising areas. The theoretical analysis of TPLSO exploration and exploitation abilities is performed and compared with several popular particle swarm optimizers. Comparative experiments on two widely used large-scale benchmark datasets demonstrate that the proposed TPLSO achieves better performance on diverse large-scale problems than several state-of-the-art algorithms.

Cardinality-constrained portfolio selection based on collaborative neurodynamic optimization

Portfolio optimization is one of the most important investment strategies in financial markets. It is practically desirable for investors, especially high-frequency traders, to consider cardinality constraints in portfolio selection, to avoid odd lots and excessive costs such as transaction fees. In this paper, a collaborative neurodynamic optimization approach is presented for cardinality-constrained portfolio selection. The expected return and investment risk in the Markowitz framework are scalarized as a weighted Chebyshev function and the cardinality constraints are equivalently represented using introduced binary variables as an upper bound. Then cardinality-constrained portfolio selection is formulated as a mixed-integer optimization problem and solved by means of collaborative neurodynamic optimization with multiple recurrent neural networks repeatedly repositioned using a particle swarm optimization rule. The distribution of resulting Pareto-optimal solutions is also iteratively perfected by optimizing the weights in the scalarized objective functions based on particle swarm optimization. Experimental results with stock data from four major world markets are discussed to substantiate the superior performance of the collaborative neurodynamic approach to several exact and metaheuristic methods.

A multi-sample particle swarm optimization algorithm based on electric field force

Aiming at the premature convergence problem of particle swarm optimization algorithm, a multi-sample particle swarm optimization (MSPSO) algorithm based on electric field force is proposed. Firstly, we introduce the concept of the electric field into the particle swarm optimization algorithm. The particles are affected by the electric field force, which makes the particles exhibit diverse behaviors. Secondly, MSPSO constructs multiple samples through two new strategies to guide particle learning. An electric field force-based comprehensive learning strategy (EFCLS) is proposed to build attractive samples and repulsive samples, thus improving search efficiency. To further enhance the convergence accuracy of the algorithm, a segment-based weighted learning strategy (SWLS) is employed to construct a global learning sample so that the particles learn more comprehensive information. In addition, the parameters of the model are adjusted adaptively to adapt to the population status in different periods. We have verified the effectiveness of these newly proposed strategies through experiments. Sixteen benchmark functions and eight well-known particle swarm optimization algorithm variants are employed to prove the superiority of MSPSO. The comparison results show that MSPSO has better performance in terms of accuracy, especially for high-dimensional spaces, while maintaining a faster convergence rate. Besides, a real-world problem also verified that MSPSO has practical application value.

A survey on evolutionary computation for complex continuous optimization

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

Evolutionary Divide-and-Conquer Algorithm for Virus Spreading Control Over Networks

The control of virus spreading over complex networks with a limited budget has attracted much attention but remains challenging. This article aims at addressing the combinatorial, discrete resource allocation problems (RAPs) in virus spreading control. To meet the challenges of increasing network scales and improve the solving efficiency, an evolutionary divide-and-conquer algorithm is proposed, namely, a coevolutionary algorithm with network-community-based decomposition (NCD-CEA). It is characterized by the community-based dividing technique and cooperative coevolution conquering thought. First, to reduce the time complexity, NCD-CEA divides a network into multiple communities by a modified community detection method such that the most relevant variables in the solution space are clustered together. The problem and the global swarm are subsequently decomposed into subproblems and subswarms with low-dimensional embeddings. Second, to obtain high-quality solutions, an alternative evolutionary approach is designed by promoting the evolution of subswarms and the global swarm, in turn, with subsolutions evaluated by local fitness functions and global solutions evaluated by a global fitness function. Extensive experiments on different networks show that NCD-CEA has a competitive performance in solving RAPs. This article advances toward controlling virus spreading over large-scale networks.

Adaptive Granularity Learning Distributed Particle Swarm Optimization for Large-Scale Optimization

Large-scale optimization has become a significant and challenging research topic in the evolutionary computation (EC) community. Although many improved EC algorithms have been proposed for large-scale optimization, the slow convergence in the huge search space and the trap into local optima among massive suboptima are still the challenges. Targeted to these two issues, this article proposes an adaptive granularity learning distributed particle swarm optimization (AGLDPSO) with the help of machine-learning techniques, including clustering analysis based on locality-sensitive hashing (LSH) and adaptive granularity control based on logistic regression (LR). In AGLDPSO, a master-slave multisubpopulation distributed model is adopted, where the entire population is divided into multiple subpopulations, and these subpopulations are co-evolved. Compared with other large-scale optimization algorithms with single population evolution or centralized mechanism, the multisubpopulation distributed co-evolution mechanism will fully exchange the evolutionary information among different subpopulations to further enhance the population diversity. Furthermore, we propose an adaptive granularity learning strategy (AGLS) based on LSH and LR. The AGLS is helpful to determine an appropriate subpopulation size to control the learning granularity of the distributed subpopulations in different evolutionary states to balance the exploration ability for escaping from massive suboptima and the exploitation ability for converging in the huge search space. The experimental results show that AGLDPSO performs better than or at least comparable with some other state-of-the-art large-scale optimization algorithms, even the winner of the competition on large-scale optimization, on all the 35 benchmark functions from both IEEE Congress on Evolutionary Computation (IEEE CEC2010) and IEEE CEC2013 large-scale optimization test suites.

A Distributed Swarm Optimizer With Adaptive Communication for Large-Scale Optimization

Large-scale optimization with high dimensionality and high computational cost becomes ubiquitous nowadays. To tackle such challenging problems efficiently, devising distributed evolutionary computation algorithms is imperative. To this end, this paper proposes a distributed swarm optimizer based on a special master-slave model. Specifically, in this distributed optimizer, the master is mainly responsible for communication with slaves, while each slave iterates a swarm to traverse the solution space. An asynchronous and adaptive communication strategy based on the request-response mechanism is especially devised to let the slaves communicate with the master efficiently. Particularly, the communication between the master and each slave is adaptively triggered during the iteration. To aid the slaves to search the space efficiently, an elite-guided learning strategy is especially designed via utilizing elite particles in the current swarm and historically best solutions found by different slaves to guide the update of particles. Together, this distributed optimizer asynchronously iterates multiple swarms to collaboratively seek the optimum in parallel. Extensive experiments on a widely used large-scale benchmark set substantiate that the distributed optimizer could: 1) achieve competitive effectiveness in terms of solution quality as compared to the state-of-the-art large-scale methods; 2) accelerate the execution of the algorithm in comparison with the sequential one and obtain almost linear speedup as the number of cores increases; and 3) preserve a good scalability to solve higher dimensional problems.

Multiple-strategy learning particle swarm optimization for large-scale optimization problems

The balance between the exploration and the exploitation plays a significant role in the meta-heuristic algorithms, especially when they are used to solve large-scale optimization problems. In this paper, we propose a multiple-strategy learning particle swarm optimization algorithm, called MSL-PSO, to solve problems with large-scale variables, in which different learning strategies are utilized in different stages. At the first stage, each individual tries to probe some positions by learning from the demonstrators who have better performance on the fitness value and the mean position of the population. All the best probed positions, each of which has the best fitness among all positions probed by its corresponding individual, will compose a new temporary population. The temporary population will be sorted on the fitness values in a descending order, and will be used for each individual to find its demonstrators, which is based on the rank of the best probed solution in the temporary population and the rank of the individual in the current population, to learn using a new strategy in the second stage. The first stage is used to improve the exploration capability, and the second one is expected to balance the convergence and diversity of the population. To verify the effectiveness of MSL-PSO for solving large-scale optimization problems, some empirical experiments are conducted, which include CEC2008 problems with 100, 500, and 1000 dimensions, and CEC2010 problems with 1000 dimensions. Experimental results show that our proposed MSL-PSO is competitive or has a better performance compared with ten state-of-the-art algorithms.

Dual-Environmental Particle Swarm Optimizer in Noisy and Noise-Free Environments

Particle swarm optimizer (PSO) is a population-based optimization technique applied to a wide range of problems. In the literature, many PSO variants have been proposed to deal with noise-free or noisy environments, respectively. While in real-life applications, noise emerges irregularly and unpredictably. As a result, PSO for a noise-free environment loses its accuracy when noise exists, while PSO for a noisy environment wastes its resampling resource when noise does not exist. To handle such scenario, a PSO variant that can work well in both noise-free and noisy environments is required, which does, to the authors' best knowledge, not exist yet. To fill such gap, this work proposes a novel PSO variant named as dual-environmental PSO (DEPSO). It uses a weighted search center based on top- k elite particles to guide the swarm. It averages their positions rather than resampling fitness values of particles to achieve noise reduction, which challenges the indispensable role of the resampling method in a noisy environment and adapts to a noise-free environment as well. Two theoretical analyses are presented for noise reduction and finer local optimization capabilities. Experimental results performed on CEC2013 benchmark functions indicate that DEPSO outperforms state-of-the-art PSO variants in both noise-free and noisy environments.

Multi-Leader Comprehensive Learning Particle Swarm Optimization with Adaptive Mutation for Economic Load Dispatch Problems

Particle swarm optimization (PSO) is one of the most popular, nature inspired optimization algorithms. The canonical PSO is easy to implement and converges fast, however, it suffers from premature convergence. The comprehensive learning particle swarm optimization (CLPSO) can achieve high exploration while it converges relatively slowly on unimodal problems. To enhance the exploitation of CLPSO without significantly impairing its exploration, a multi-leader (ML) strategy is combined with CLPSO. In ML strategy, a group of top ranked particles act as the leaders to guide the motion of the whole swarm. Each particle is randomly assigned with an individual leader and the leader is refreshed dynamically during the optimization process. To activate the stagnated particles, an adaptive mutation (AM) strategy is introduced. Combining the ML and the AM strategies with CLPSO simultaneously, the resultant algorithm is referred to as multi-leader comprehensive learning particle swarm optimization with adaptive mutation (ML-CLPSO-AM). To evaluate the performance of ML-CLPSO-AM, the CEC2017 test suite was employed. The test results indicate that ML-CLPSO-AM performs better than ten popular PSO variants and six other types of representative evolutionary algorithms and meta-heuristics. To validate the effectiveness of ML-CLPSO-AM in real-life applications, ML-CLPSO-AM was applied to economic load dispatch (ELD) problems.

Set-Based Discrete Particle Swarm Optimization Based on Decomposition for Permutation-Based Multiobjective Combinatorial Optimization Problems

This paper studies a specific class of multiobjective combinatorial optimization problems (MOCOPs), namely the permutation-based MOCOPs. Many commonly seen MOCOPs, e.g., multiobjective traveling salesman problem (MOTSP), multiobjective project scheduling problem (MOPSP), belong to this problem class and they can be very different. However, as the permutation-based MOCOPs share the inherent similarity that the structure of their search space is usually in the shape of a permutation tree, this paper proposes a generic multiobjective set-based particle swarm optimization methodology based on decomposition, termed MS-PSO/D. In order to coordinate with the property of permutation-based MOCOPs, MS-PSO/D utilizes an element-based representation and a constructive approach. Through this, feasible solutions under constraints can be generated step by step following the permutation-tree-shaped structure. And problem-related heuristic information is introduced in the constructive approach for efficiency. In order to address the multiobjective optimization issues, the decomposition strategy is employed, in which the problem is converted into multiple single-objective subproblems according to a set of weight vectors. Besides, a flexible mechanism for diversity control is provided in MS-PSO/D. Extensive experiments have been conducted to study MS-PSO/D on two permutation-based MOCOPs, namely the MOTSP and the MOPSP. Experimental results validate that the proposed methodology is promising.

Distributed Differential Evolution Based on Adaptive Mergence and Split for Large-Scale Optimization

Nowadays, large-scale optimization problems are ubiquitous in many research fields. To deal with such problems efficiently, this paper proposes a distributed differential evolution with adaptive mergence and split (DDE-AMS) on subpopulations. The novel mergence and split operators are designed to make full use of limited population resource, which is important for large-scale optimization. They are adaptively performed based on the performance of the subpopulations. During the evolution, once a subpopulation finds a promising region, the current worst performing subpopulation will merge into it. If the merged subpopulation could not continuously provide competitive solutions, it will be split in half. In this way, the number of subpopulations is adaptively adjusted and better performing subpopulations obtain more individuals. Thus, population resource can be adaptively arranged for subpopulations during the evolution. Moreover, the proposed algorithm is implemented with a parallel master-slave manner. Extensive experiments are conducted on 20 widely used large-scale benchmark functions. Experimental results demonstrate that the proposed DDE-AMS could achieve competitive or even better performance compared with several state-of-the-art algorithms. The effects of DDE-AMS components, adaptive behavior, scalability, and parameter sensitivity are also studied. Finally, we investigate the speedup ratios of DDE-AMS with different computation resources.

Cooperative Hierarchical PSO With Two Stage Variable Interaction Reconstruction for Large Scale Optimization

Large scale optimization problems arise in diverse fields. Decomposing the large scale problem into small scale subproblems regarding the variable interactions and optimizing them cooperatively are critical steps in an optimization algorithm. To explore the variable interactions and perform the problem decomposition tasks, we develop a two stage variable interaction reconstruction algorithm. A learning model is proposed to explore part of the variable interactions as prior knowledge. A marginalized denoising model is proposed to construct the overall variable interactions using the prior knowledge, with which the problem is decomposed into small scale modules. To optimize the subproblems and relieve premature convergence, we propose a cooperative hierarchical particle swarm optimization framework, where the operators of contingency leadership, interactional cognition, and self-directed exploitation are designed. Finally, we conduct theoretical analysis for further understanding of the proposed algorithm. The analysis shows that the proposed algorithm can guarantee converging to the global optimal solutions if the problems are correctly decomposed. Experiments are conducted on the CEC2008 and CEC2010 benchmarks. The results demonstrate the effectiveness, convergence, and usefulness of the proposed algorithm.

Benchmarking Stochastic Algorithms for Global Optimization Problems by Visualizing Confidence Intervals

The popular performance profiles and data profiles for benchmarking deterministic optimization algorithms are extended to benchmark stochastic algorithms for global optimization problems. A general confidence interval is employed to replace the significance test, which is popular in traditional benchmarking methods but suffering more and more criticisms. Through computing confidence bounds of the general confidence interval and visualizing them with performance profiles and (or) data profiles, our benchmarking method can be used to compare stochastic optimization algorithms by graphs. Compared with traditional benchmarking methods, our method is synthetic statistically and therefore is suitable for large sets of benchmark problems. Compared with some sample-mean-based benchmarking methods, e.g., the method adopted in black-box-optimization-benchmarking workshop/competition, our method considers not only sample means but also sample variances. The most important property of our method is that it is a distribution-free method, i.e., it does not depend on any distribution assumption of the population. This makes it a promising benchmarking method for stochastic optimization algorithms. Some examples are provided to illustrate how to use our method to compare stochastic optimization algorithms.