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406:(called particles). These particles are moved around in the search-space according to a few simple formulae. The movements of the particles are guided by their own best-known position in the search-space as well as the entire swarm's best-known position. When improved positions are being discovered these will then come to guide the movements of the swarm. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered.
1055:, remain constant throughout the optimization process. However, it was shown that these simplifications do not affect the boundaries found by these studies for parameter where the swarm is convergent. Considerable effort has been made in recent years to weaken the modeling assumption utilized during the stability analysis of PSO, with the most recent generalized result applying to numerous PSO variants and utilized what was shown to be the minimal necessary modeling assumptions.
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with a higher convergence speed. It enables automatic control of the inertia weight, acceleration coefficients, and other algorithmic parameters at the run time, thereby improving the search effectiveness and efficiency at the same time. Also, APSO can act on the globally best particle to jump out of the likely local optima. However, APSO will introduce new algorithm parameters, it does not introduce additional design or implementation complexity nonetheless.
1162:(another popular metaheuristic) but it was later found to be defective as it was strongly biased in its optimization search towards similar values for different dimensions in the search space, which happened to be the optimum of the benchmark problems considered. This bias was because of a programming error, and has now been fixed.
1073:, so as to form a leading converging exemplar and to be effective with any PSO topology. The aims are to improve the performance of PSO overall, including faster global convergence, higher solution quality, and stronger robustness. However, such studies do not provide theoretical evidence to actually prove their claims.
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As the PSO equations given above work on real numbers, a commonly used method to solve discrete problems is to map the discrete search space to a continuous domain, to apply a classical PSO, and then to demap the result. Such a mapping can be very simple (for example by just using rounded values) or
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Another school of thought is that the behaviour of a PSO swarm is not well understood in terms of how it affects actual optimization performance, especially for higher-dimensional search-spaces and optimization problems that may be discontinuous, noisy, and time-varying. This school of thought merely
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The topology of the swarm defines the subset of particles with which each particle can exchange information. The basic version of the algorithm uses the global topology as the swarm communication structure. This topology allows all particles to communicate with all the other particles, thus the whole
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New and more sophisticated PSO variants are also continually being introduced in an attempt to improve optimization performance. There are certain trends in that research; one is to make a hybrid optimization method using PSO combined with other optimizers, e.g., combined PSO with biogeography-based
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Without the need for a trade-off between convergence ('exploitation') and divergence ('exploration'), an adaptive mechanism can be introduced. Adaptive particle swarm optimization (APSO) features better search efficiency than standard PSO. APSO can perform global search over the entire search space
2011:
real numbers, and these operators are simply -, *, +, and again +. But all these mathematical objects can be defined in a completely different way, in order to cope with binary problems (or more generally discrete ones), or even combinatorial ones. One approach is to redefine the operators based on
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A common belief amongst researchers is that the swarm behaviour varies between exploratory behaviour, that is, searching a broader region of the search-space, and exploitative behaviour, that is, a locally oriented search so as to get closer to a (possibly local) optimum. This school of thought has
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Convergence of the sequence of solutions has been investigated for PSO. These analyses have resulted in guidelines for selecting PSO parameters that are believed to cause convergence to a point and prevent divergence of the swarm's particles (particles do not move unboundedly and will converge to
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from a single particle. However, this approach might lead the swarm to be trapped into a local minimum, thus different topologies have been used to control the flow of information among particles. For instance, in local topologies, particles only share information with a subset of particles. This
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A commonly used swarm topology is the ring, in which each particle has just two neighbours, but there are many others. The topology is not necessarily static. In fact, since the topology is related to the diversity of communication of the particles, some efforts have been done to create adaptive
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A series of standard implementations have been created by leading researchers, "intended for use both as a baseline for performance testing of improvements to the technique, as well as to represent PSO to the wider optimization community. Having a well-known, strictly-defined standard algorithm
333:. Each particle's movement is influenced by its local best known position, but is also guided toward the best known positions in the search-space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions.
1143:. Simplifying PSO was originally suggested by Kennedy and has been studied more extensively, where it appeared that optimization performance was improved, and the parameters were easier to tune and they performed more consistently across different optimization problems.
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To prevent divergence ("explosion") the inertia weight must be smaller than 1. The two other parameters can be then derived thanks to the constriction approach, or freely selected, but the analyses suggest convergence domains to constrain them. Typical values are in
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somewhere). However, the analyses were criticized by
Pedersen for being oversimplified as they assume the swarm has only one particle, that it does not use stochastic variables and that the points of attraction, that is, the particle's best known position
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Another research trend is to try to alleviate premature convergence (that is, optimization stagnation), e.g. by reversing or perturbing the movement of the PSO particles, another approach to deal with premature convergence is the use of multiple swarms
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tries to find PSO algorithms and parameters that cause good performance regardless of how the swarm behaviour can be interpreted in relation to e.g. exploration and exploitation. Such studies have led to the simplification of the PSO algorithm, see
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Numerous variants of even a basic PSO algorithm are possible. For example, there are different ways to initialize the particles and velocities (e.g. start with zero velocities instead), how to dampen the velocity, only update
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Another simpler variant is the accelerated particle swarm optimization (APSO), which also does not need to use velocity and can speed up the convergence in many applications. A simple demo code of APSO is available.
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Almasi, O. N. and
Khooban, M. H. (2017). A parsimonious SVM model selection criterion for classification of real-world data sets via an adaptive population-based algorithm. Neural Computing and Applications, 1-9.
4000:
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Nobile, M.; Besozzi, D.; Cazzaniga, P.; Mauri, G.; Pescini, D. (2012). "A GPU-Based Multi-Swarm PSO Method for
Parameter Estimation in Stochastic Biological Systems Exploiting Discrete-Time Target Series".
963:
nearest particles" β or, more often, a social one, i.e. a set of particles that is not depending on any distance. In such cases, the PSO variant is said to be local best (vs global best for the basic PSO).
987:
been prevalent since the inception of PSO. This school of thought contends that the PSO algorithm and its parameters must be chosen so as to properly balance between exploration and exploitation to avoid
1908:
1131:). The multi-swarm approach can also be used to implement multi-objective optimization. Finally, there are developments in adapting the behavioural parameters of PSO during optimization.
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results. One attempt at addressing this issue is the development of an "orthogonal learning" strategy for an improved use of the information already existing in the relationship between
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The choice of PSO parameters can have a large impact on optimization performance. Selecting PSO parameters that yield good performance has therefore been the subject of much research.
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Jarboui, B.; Damak, N.; Siarry, P.; Rebai, A. (2008). "A combinatorial particle swarm optimization for solving multi-mode resource-constrained project scheduling problems".
3607:"Pathological Brain Detection in Magnetic Resonance Imaging Scanning by Wavelet Entropy and Hybridization of Biogeography-based Optimization and Particle Swarm Optimization"
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Nobile, M.S; Pasi, G.; Cazzaniga, P.; Besozzi, D.; Colombo, R.; Mauri, G. (2015). "Proactive particles in swarm optimization: a self-tuning algorithm based on fuzzy logic".
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In this variant of PSO one dispences with both the particle's velocity and the particle's best position. The particle position is updated according to the following rule,
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Mason, Karl; Duggan, Jim; Howley, Enda (2018). "A Meta
Optimisation Analysis of Particle Swarm Optimisation Velocity Update Equations for Watershed Management Learning".
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Initialization of velocities may require extra inputs. The Bare Bones PSO variant has been proposed in 2003 by James
Kennedy, and does not need to use velocity at all.
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provides a valuable point of comparison which can be used throughout the field of research to better test new advances." The latest is
Standard PSO 2011 (SPSO-2011).
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363:. The algorithm was simplified and it was observed to be performing optimization. The book by Kennedy and Eberhart describes many philosophical aspects of PSO and
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Besides, through the utilization of a scale-adaptive fitness evaluation mechanism, PSO can efficiently address computationally expensive optimization problems.
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Nobile, M.S; Cazzaniga, P.; Besozzi, D.; Colombo, R.; Mauri, G.; Pasi, G. (2018). "Fuzzy Self-Tuning PSO: a settings-free algorithm for global optimization".
3704:
Cheung, N. J.; Ding, X.-M.; Shen, H.-B. (2013). "OptiFel: A Convergent
Heterogeneous Particle Sarm Optimization Algorithm for Takagi-Sugeno Fuzzy Modeling".
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Cleghorn, Christopher W.; Engelbrecht, Andries. (2018). "Particle Swarm
Stability: A Theoretical Extension using the Non-Stagnate Distribution Assumption".
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In this variant of PSO one dispences with the velocity of the particles and instead updates the positions of the particles using the following simple rule,
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The termination criterion can be the number of iterations performed, or a solution where the adequate objective function value is found. The parameters w, Ο
4013:
3467:"Scale adaptive fitness evaluation-based particle swarm optimisation for hyperparameter and architecture optimisation in neural networks and deep learning"
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Convergence to a local optimum has been analyzed for PSO in and. It has been proven that PSO needs some modification to guarantee finding a local optimum.
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Another school of thought is that PSO should be simplified as much as possible without impairing its performance; a general concept often referred to as
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as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. Also, PSO does not use the
4805:
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Zambrano-Bigiarini, M.; Clerc, M.; Rojas, R. (2013). "Standard
Particle Swarm Optimisation 2011 at CEC-2013: A baseline for future PSO improvements".
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Cazzaniga, P.; Nobile, M.S.; Besozzi, D. (2015). "The impact of particles initialization in PSO: parameter estimation as a case in point, (Canada)".
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after the entire swarm has been updated, etc. Some of these choices and their possible performance impact have been discussed in the literature.
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Mason, Karl; Duggan, Jim; Howley, Enda (2017). "Multi-objective dynamic economic emission dispatch using particle swarm optimisation variants".
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and this increases the risk of making errors in its description and implementation. A good example of this presented a promising variant of a
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Performance landscape showing how a simple PSO variant performs in aggregate on several benchmark problems when varying two PSO parameters.
4150:
Liu, Yang (2009). "Automatic calibration of a rainfallβrunoff model using a fast and elitist multi-objective particle swarm algorithm".
413:: β β β be the cost function which must be minimized. The function takes a candidate solution as an argument in the form of a
3312:
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Oliveira, M.; Pinheiro, D.; Andrade, B.; Bastos-Filho, C.; Menezes, R. (2016). "Communication
Diversity in Particle Swarm Optimizers".
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represent the lower and upper boundaries of the search-space respectively. The w parameter is the inertia weight. The parameters Ο
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Clerc, M.; Kennedy, J. (2002). "The particle swarm - explosion, stability, and convergence in a multidimensional complex space".
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by doing computational experiments on a finite number of optimization problems. This means a metaheuristic such as PSO cannot be
99:
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Bonyadi, M. R.; Michalewicz, Z. (2017). "Particle swarm optimization for single objective continuous space problems: a review".
4674:
235:
371:. In 2017, a comprehensive review on theoretical and experimental works on PSO has been published by Bonyadi and Michalewicz.
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Chen, Wei-neng; Zhang, Jun (2010). "A novel set-based particle swarm optimization method for discrete optimization problem".
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Taherkhani, M.; Safabakhsh, R. (2016). "A novel stability-based adaptive inertia weight for particle swarm optimization".
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with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, here dubbed
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into account when moving the PSO particles and non-dominated solutions are stored so as to approximate the pareto front.
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By using the ring topology, PSO can attain generation-level parallelism, significantly enhancing the evolutionary speed.
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and produces a real number as output which indicates the objective function value of the given candidate solution. The
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Tu, Z.; Lu, Y. (2008). "Corrections to "A Robust Stochastic Genetic Algorithm (StGA) for Global Numerical Optimization
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This means that determining the convergence capabilities of different PSO algorithms and parameters still depends on
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1325:{\displaystyle {\vec {x}}_{i}=G\left({\frac {{\vec {p}}_{i}+{\vec {g}}}{2}},||{\vec {p}}_{i}-{\vec {g}}||\right)\,,}
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3161:"Generation-Level Parallelism for Evolutionary Computation: A Pipeline-Based Parallel Particle Swarm Optimization"
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Niknam, T.; Amiri, B. (2010). "An efficient hybrid approach based on PSO, ACO and k-means for cluster analysis".
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2734:
2021:
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Trelea, I.C. (2003). "The Particle Swarm Optimization Algorithm: convergence analysis and parameter selection".
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Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics. Lecture Notes in Computer Science
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Cleghorn, Christopher W (2014). "Particle Swarm Convergence: Standardized Analysis and Topological Influence".
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Clerc, M. (2006). Particle Swarm Optimization. ISTE (International Scientific and Technical Encyclopedia), 2006
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Accelerated particle swarm optimization and support vector machine for business optimization and applications
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be the best known position of the entire swarm. A basic PSO algorithm to minimize the cost function is then:
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1031:) in which all particles have converged to a point in the search-space, which may or may not be the optimum,
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Proceedings of the 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2015), Istanbul (Turkey)
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Tu, Z.; Lu, Y. (2004). "A robust stochastic genetic algorithm (StGA) for global numerical optimization".
2725:. University of Southampton, School of Engineering Sciences, Computational Engineering and Design Group.
1990:
However, it can be noted that the equations of movement make use of operators that perform four actions:
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240:
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Proceedings of IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology
2664:"Optimized Particle Swarm Optimization (OPSO) and its application to artificial neural network training"
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1717:{\displaystyle {\vec {x}}_{i}\leftarrow (1-\beta ){\vec {x}}_{i}+\beta {\vec {g}}+\alpha L{\vec {u}}\,,}
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4252:
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Parsopoulos, K.; Vrahatis, M. (2002). "Particle swarm optimization method in multiobjective problems".
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Clerc, M. (2005). Binary Particle Swarm Optimisers: toolbox, derivations, and mathematical insights,
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A Novel Particle Swarm Optimization Algorithm for Multi-Objective Combinatorial Optimization Problem
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computing the difference of two positions. The result is a velocity (more precisely a displacement)
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of the problem being optimized, which means PSO does not require that the optimization problem be
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4016:, Conference on Systems, Man, and Cybernetics, Piscataway, NJ: IEEE Service Center, pp. 4104-4109
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Bratton, Daniel; Kennedy, James (2007). "Defining a Standard for Particle Swarm Optimization".
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The PSO parameters can also be tuned by using another overlaying optimizer, a concept known as
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2637:. The University of Texas - Pan American, Department of Electrical Engineering. Archived from
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are selected by the practitioner and control the behaviour and efficacy of the PSO method (
394:. However, metaheuristics such as PSO do not guarantee an optimal solution is ever found.
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Pedersen, M.E.H.; Chipperfield, A.J. (2010). "Simplifying particle swarm optimization".
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An Automatic Regrouping Mechanism to Deal with Stagnation in Particle Swarm Optimization
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Xinchao, Z. (2010). "A perturbed particle swarm algorithm for numerical optimization".
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Wang, Ye-Qun; Li, Jian-Yu; Chen, Chun-Hua; Zhang, Jun; Zhan, Zhi-Hui (September 2023).
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Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600)
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Kennedy, J.; Mendes, R. (2002). "Population structure and particle swarm performance".
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Shi, Y.; Eberhart, R.C. (1998). "Parameter selection in particle swarm optimization".
2298:"A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications"
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A basic variant of the PSO algorithm works by having a population (called a swarm) of
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Discrete Particle Swarm Optimization, illustrated by the Traveling Salesman Problem
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Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS'03 (Cat. No.03EX706)
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3335:"A locally convergent rotationally invariant particle swarm optimization algorithm"
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2443:"Comparing inertia weights and constriction factors in particle swarm optimization"
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is a repository for information on PSO. Several source codes are freely available.
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2504:(PhD thesis). University of Pretoria, Faculty of Natural and Agricultural Science.
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4003:. 'International Journal of Applied Metaheuristic Computing (IJAMC)', 2(4), 41-57
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2262:"Analysis of the publications on the applications of particle swarm optimisation"
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are the parameters of the method. As a refinement of the method one can decrease
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3502:. Evolutionary Computation (CEC), 2013 IEEE Congress on. pp. 2337β2344.
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Stochastic Star Communication Topology in Evolutionary Particle Swarms (EPSO)
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942:, or even fine-tuned during the optimization, e.g., by means of fuzzy logic.
539:) Initialize the particle's best known position to its initial position:
375:
368:
197:
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Liu, Q (2015). "Order-2 stability analysis of particle swarm optimization".
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Pedersen, M.E.H. (2010). "Good parameters for particle swarm optimization".
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3113:. International Journal of Swarm Intelligence Research (IJSIR), 2(2), 22-41
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Kennedy, J. (1997). "The particle swarm: social adaptation of knowledge".
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2226:"An analysis of publications on particle swarm optimisation applications"
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topologies (SPSO, APSO, stochastic star, TRIBES, Cyber Swarm, and C-PSO)
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Proceedings of IEEE International Conference on Evolutionary Computation
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Proceedings of IEEE International Conference on Evolutionary Computation
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3194:. Lecture Notes in Computer Science. Vol. 8667. pp. 134β145.
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Population Topologies and Their Influence in Particle Swarm Performance
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Shi, Y.; Eberhart, R.C. (1998). "A modified particle swarm optimizer".
999:
to the optimum. This belief is the precursor of many PSO variants, see
414:
3956:"MOPSO: A Proposal for Multiple Objective Particle Swarm Optimization"
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355:, as a stylized representation of the movement of organisms in a bird
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3640:"Extending Particle Swarm Optimisers with Self-Organized Criticality"
3026:. Lecture Notes in Computer Science. Vol. 9882. pp. 77β88.
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optimization, and the incorporation of an effective learning method.
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Proceedings of the Fourth Congress on Evolutionary Computation (CEC)
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Parameters have also been tuned for various optimization scenarios.
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4029:, New Optimization Techniques in Engineering, Springer, pp. 219-239
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Convergence of the sequence of solutions (aka, stability analysis,
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330:
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Simulation of PSO convergence in a two-dimensional space (Matlab).
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DEPSO: hybrid particle swarm with differential evolution operator
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Kennedy, J.; Eberhart, R. (1995). "Particle Swarm Optimization".
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be the number of particles in the swarm, each having a position
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Proceedings of IEEE International Conference on Neural Networks
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are often called cognitive coefficient and social coefficient.
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IEEE International Conference on Systems, Man, and Cybernetics
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Proceedings of Parallel Problem Solving from Nature VII (PPSN)
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as to why and how the PSO algorithm can perform optimization.
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Proceedings of the ACM Symposium on Applied Computing (SAC)
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A discrete binary version of the particle swarm algorithm
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of particle swarms optimizing three benchmark functions.
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Convergence to a local optimum where all personal bests
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3918:"Search Results: APSO - File Exchange - MATLAB Central"
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Xie, Xiao-Feng; Zhang, Wen-Jun; Yang, Zhi-Lian (2002).
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Yin, P., Glover, F., Laguna, M., & Zhu, J. (2011).
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Proceedings of the Particle Swarm Optimization Workshop
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Proceedings of the Congress on Evolutionary Computation
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are the position and the best position of the particle
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as is required by classic optimization methods such as
3491:
3313:"A convergence proof for the particle swarm optimizer"
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2587:
2585:
1981:
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Another argument in favour of simplifying PSO is that
367:. An extensive survey of PSO applications is made by
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Kennedy, James (2003). "Bare bones particle swarms".
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2389:
2007:
Usually a position and a velocity are represented by
1936:
1916:
1866:
1846:
1814:
1782:
1762:
1733:
1614:
1559:
1539:
1510:
1462:
1433:
1413:
1377:
1341:
1182:
909:
3415:
Zhan, Z-H.; Zhang, J.; Li, Y; Chung, H.S-H. (2009).
3010:
Particle swarm optimiser with neighbourhood operator
2395:
4205:
3947:
3930:
3631:
3532:
3265:
3122:
2711:
2709:
2655:
2582:
2557:
2555:
2517:
2515:
2513:
2511:
2434:
2171:
2169:
2167:
1974:, in which the objective function comparison takes
1038:or, alternatively, the swarm's best known position
959:subset can be a geometrical one β for example "the
4806:Task allocation and partitioning of social insects
3653:
3424:IEEE Transactions on Systems, Man, and Cybernetics
3332:
2899:
2662:Meissner, M.; Schmuker, M.; Schneider, G. (2006).
2428:Proceedings of Evolutionary Programming VII (EP98)
2078:
1954:
1922:
1903:{\displaystyle \alpha _{n}=\alpha _{0}\gamma ^{n}}
1902:
1852:
1832:
1800:
1776:is the typical length of the problem at hand, and
1768:
1748:
1716:
1585:
1545:
1525:
1492:
1448:
1419:
1399:
1363:
1324:
1121:
927:
683:(0,1) Update the particle's velocity:
3554:
3410:
3408:
3380:"Orthogonal Learning Particle Swarm Optimization"
3333:Bonyadi, Mohammad reza.; Michalewicz, Z. (2014).
2719:Tuning & Simplifying Heuristical Optimization
2421:
2419:
2074:
2072:
1997:multiplying a velocity by a numerical coefficient
4818:
3907:, NDT 2011, Springer CCIS 136, pp. 53-66 (2011).
3414:
3378:Zhan, Z-H.; Zhang, J.; Li, Y; Shi, Y-H. (2011).
3134:. Honolulu, HI. pp. 289β296. Archived from
3132:IEEE Swarm Intelligence Symposium 2007 (SIS2007)
2706:
2626:
2624:
2601:Journal of Artificial Evolution and Applications
2591:
2552:
2508:
2490:
2488:
2269:Journal of Artificial Evolution and Applications
2164:
2115:
948:
3960:Congress on Evolutionary Computation (CEC'2002)
3954:Coello Coello, C.; Salazar Lechuga, M. (2002).
3816:
3464:
3087:Miranda, V., Keko, H. and Duque, Γ. J. (2008).
2198:
2143:
2141:
2121:
1965:
1023:typically refers to two different definitions:
3999:Roy, R., Dehuri, S., & Cho, S. B. (2012).
3703:
3500:2013 IEEE Congress on Evolutionary Computation
3405:
3377:
3373:
3371:
2462:
2416:
2349:
2345:
2343:
2341:
2339:
2337:
2069:
4304:
4191:
4080:IEEE Transactions on Evolutionary Computation
3827:IEEE Transactions on Evolutionary Computation
3790:IEEE Transactions on Evolutionary Computation
3637:
3538:
3387:IEEE Transactions on Evolutionary Computation
2930:
2876:
2818:
2816:
2786:
2621:
2524:IEEE Transactions on Evolutionary Computation
2494:
2485:
2253:
2217:
2192:
471: β β in the search-space and a velocity
429:is not known. The goal is to find a solution
270:
3966:
3756:
3471:CAAI Transactions on Intelligence Technology
3259:
3123:Elshamy, W.; Rashad, H.; Bahgat, A. (2007).
3116:
2521:
2138:
3560:
3368:
3326:
2440:
2425:
2334:
2147:
794:Update the particle's best known position:
317:, and moving these particles around in the
4311:
4297:
4198:
4184:
4071:
4012:Kennedy, J. & Eberhart, R. C. (1997).
3781:
3732:
2853:
2813:
2312:
1756:is a random uniformly distributed vector,
1150:can only have their efficacy demonstrated
745:) Update the particle's position:
510:Initialize the particle's position with a
277:
263:
4091:
4077:
3688:A dissipative particle swarm optimization
3680:
3622:
3604:
3581:
3482:
3310:
3287:
3125:"Clubs-based Particle Swarm Optimization"
3077:https://doi.org/10.1007/s00521-017-2930-y
2940:
2935:. Vol. 2. pp. 1671β1676 vol.2.
2915:
2844:
2769:
2689:
2679:
2612:
2535:
2280:
1710:
1318:
3765:Nature-Inspired Metaheuristic Algorithms
3189:
2905:
2715:
2498:An Analysis of Particle Swarm Optimizers
890:
880:
833:Update the swarm's best known position:
578:update the swarm's best known position:
20:
3862:
3659:
3595:(SMCC), Washington, DC, USA: 3816-3821.
3587:Zhang, Wen-Jun; Xie, Xiao-Feng (2003).
3216:
3158:
2175:
1597:Accelerated Particle Swarm Optimization
485:be the best known position of particle
4819:
3417:"Adaptive Particle Swarm Optimization"
2561:
2353:2007 IEEE Swarm Intelligence Symposium
2321:"Standard Particle Swarm Optimisation"
1076:
886:
453:in the search-space, which would mean
4414:Patterns of self-organization in ants
4292:
4179:
3611:Progress in Electromagnetics Research
3111:A Complementary Cyber Swarm Algorithm
2630:
2318:
2295:
1493:{\displaystyle G({\vec {x}},\sigma )}
3822:
3787:
3762:
3694:(CEC), Honolulu, HI, USA: 1456-1461.
3692:Congress on Evolutionary Computation
2999:(PhD thesis). Universidade do Minho.
2302:Mathematical Problems in Engineering
2259:
2223:
2199:Kennedy, J.; Eberhart, R.C. (2001).
2126:. Vol. IV. pp. 1942β1948.
1051:and the swarm's best known position
1008:
589:Initialize the particle's velocity:
105:Evolutionary multimodal optimization
4149:
4053:Applied Mathematics and Computation
3222:
2592:Bratton, D.; Blackwell, T. (2008).
1982:Binary, discrete, and combinatorial
1962:is the decrease control parameter.
1930:is the number of the iteration and
1833:{\displaystyle \alpha \sim 0.1-0.5}
954:swarm share the same best position
632:a termination criterion is not met
25:A particle swarm searching for the
13:
4424:symmetry breaking of escaping ants
3744:. Vol. 7264. pp. 74β85.
3706:IEEE Transactions on Fuzzy Systems
3649:. Vol. 2. pp. 1588β1593.
2825:Swarm and Evolutionary Computation
1801:{\displaystyle \beta \sim 0.1-0.7}
1134:
14:
4843:
4258:Infinite-dimensional optimization
4120:
2463:Carlisle, A.; Dozier, G. (2001).
2003:applying a velocity to a position
1955:{\displaystyle 0<\gamma <1}
1165:
974:
301:) is a computational method that
4461:
4152:Expert Systems with Applications
3903:X. S. Yang, S. Deb and S. Fong,
3638:Lovbjerg, M.; Krink, T. (2002).
3539:Lovbjerg, M.; Krink, T. (2002).
3165:IEEE Transactions on Cybernetics
2441:Eberhart, R.C.; Shi, Y. (2000).
1593:signifies the norm of a vector.
1112:
995:yet still ensure a good rate of
336:PSO is originally attributed to
130:Promoter based genetic algorithm
4207:Major subfields of optimization
4044:
4032:
4019:
4006:
3993:
3910:
3897:
3856:
3697:
3598:
3458:
3304:
3183:
3152:
3103:
3094:
3081:
3068:
3056:
3002:
2989:
2022:Artificial bee colony algorithm
1122:Alleviate premature convergence
658: = 1, ...,
644: = 1, ...,
503: = 1, ...,
65:Cellular evolutionary algorithm
2870:10.1109/FUZZ-IEEE.2015.7337957
2594:"A Simplified Recombinant PSO"
2564:Information Processing Letters
2474:. pp. 1β6. Archived from
2449:. Vol. 1. pp. 84β88.
2289:
1740:
1704:
1683:
1659:
1649:
1637:
1634:
1622:
1579:
1574:
1566:
1561:
1517:
1487:
1475:
1466:
1440:
1400:{\displaystyle {\vec {p}}_{i}}
1385:
1364:{\displaystyle {\vec {x}}_{i}}
1349:
1309:
1304:
1297:
1276:
1265:
1260:
1244:
1223:
1190:
1014:
922:
910:
1:
2981:: CS1 maint: date and year (
2576:10.1016/S0020-0190(02)00447-7
2062:
2057:Dispersive flies optimisation
1970:PSO has also been applied to
1456:is the global best position;
1000:
949:Neighbourhoods and topologies
216:Cartesian genetic programming
135:Spiral optimization algorithm
4385:Mixed-species foraging flock
4336:Agent-based model in biology
4318:
3987:10.1016/j.neucom.2017.03.086
3605:Zhang, Y.; Wang, S. (2015).
3200:10.1007/978-3-319-09952-1_12
2032:Derivative-free optimization
1966:Multi-objective optimization
1019:In relation to PSO the word
397:
231:Multi expression programming
7:
4632:Particle swarm optimization
4273:Multiobjective optimization
3750:10.1007/978-3-642-29066-4_7
3032:10.1007/978-3-319-44427-7_7
3008:Suganthan, Ponnuthurai N. "
2837:10.1016/j.swevo.2017.09.001
2015:
1088:
348:and was first intended for
295:particle swarm optimization
110:Particle swarm optimization
18:Iterative simulation method
10:
4848:
4341:Collective animal behavior
4253:Combinatorial optimization
4164:10.1016/j.eswa.2008.10.086
3718:10.1109/TFUZZ.2013.2278972
3674:10.1016/j.asoc.2009.06.010
3575:10.1016/j.asoc.2009.07.001
3436:10.1109/TSMCB.2009.2015956
2893:10.1109/CIBCB.2015.7300288
2807:10.1016/j.asoc.2017.10.018
2780:10.1016/j.asoc.2009.08.029
2495:van den Bergh, F. (2001).
2410:10.1016/j.asoc.2015.10.004
1749:{\displaystyle {\vec {u}}}
1586:{\displaystyle ||\dots ||}
1526:{\displaystyle {\vec {x}}}
1449:{\displaystyle {\vec {g}}}
221:Linear genetic programming
168:Clonal selection algorithm
120:Natural evolution strategy
4733:
4695:
4650:
4602:
4470:
4459:
4326:
4213:
4102:10.1109/tevc.2009.2030331
4065:10.1016/j.amc.2007.04.096
3399:10.1109/TEVC.2010.2052054
3354:10.1007/s11721-014-0095-1
3280:10.1007/s11721-017-0141-x
3177:10.1109/TCYB.2020.3028070
2716:Pedersen, M.E.H. (2010).
4670:Self-propelled particles
4171:Links to PSO source code
3873:10.1109/SIS.2003.1202251
3839:10.1109/TEVC.2008.926734
3802:10.1109/TEVC.2004.831258
3508:10.1109/CEC.2013.6557848
3225:Evolutionary Computation
2951:10.1109/CEC.2002.1004493
2233:Technical Report CSM-469
2186:10.1109/ICEC.1997.592326
2158:10.1109/ICEC.1998.699146
2132:10.1109/ICNN.1995.488968
2081:Evolutionary Computation
2037:Multi-swarm optimization
1972:multi-objective problems
1129:multi-swarm optimization
85:Evolutionary computation
4832:Evolutionary algorithms
4751:Collective intelligence
4617:Ant colony optimization
4268:Constraint satisfaction
3320:Fundamenta Informaticae
2908:Technical Report HL1001
2681:10.1186/1471-2105-7-125
2365:10.1109/SIS.2007.368035
2328:HAL Open Access Archive
1853:{\displaystyle \alpha }
1546:{\displaystyle \sigma }
1533:and standard deviation
457:is the global minimum.
4771:Microbial intelligence
4431:Shoaling and schooling
4243:Stochastic programming
4223:Fractional programming
4127:Particle Swarm Central
3662:Applied Soft Computing
3563:Applied Soft Computing
3064:Particle Swarm Central
2795:Applied Soft Computing
2758:Applied Soft Computing
2465:"An Off-The-Shelf PSO"
2398:Applied Soft Computing
1956:
1924:
1904:
1854:
1834:
1802:
1770:
1750:
1718:
1587:
1547:
1527:
1494:
1450:
1421:
1401:
1365:
1326:
929:
896:
75:Differential evolution
55:Artificial development
46:Evolutionary algorithm
30:
4238:Nonlinear programming
4233:Quadratic programming
3962:. pp. 1051β1056.
3941:10.1145/508791.508907
2000:adding two velocities
1957:
1925:
1905:
1860:with each iteration,
1855:
1835:
1803:
1771:
1751:
1719:
1588:
1548:
1528:
1495:
1451:
1422:
1402:
1366:
1327:
989:premature convergence
930:
894:
665:Pick random numbers:
512:uniformly distributed
323:mathematical formulae
291:computational science
226:Grammatical evolution
188:Genetic fuzzy systems
24:
4791:Spatial organization
4756:Decentralised system
4594:Sea turtle migration
4448:Swarming (honey bee)
3935:. pp. 603β607.
3624:10.2528/pier15040602
3237:10.1162/EVCO_a_00129
3159:Jian-Yu, Li (2021).
2359:. pp. 120β127.
2180:. pp. 303β308.
2093:10.1162/EVCO_r_00180
1987:more sophisticated.
1934:
1914:
1864:
1844:
1812:
1780:
1760:
1731:
1612:
1557:
1537:
1508:
1460:
1431:
1411:
1375:
1339:
1180:
907:
392:quasi-newton methods
325:over the particle's
321:according to simple
309:trying to improve a
4766:Group size measures
4328:Biological swarming
4278:Simulated annealing
4248:Robust optimization
4228:Integer programming
3763:Yang, X.S. (2008).
3550:. pp. 621β630.
2995:Mendes, R. (2004).
2614:10.1155/2008/654184
2546:10.1109/4235.985692
2430:. pp. 591β600.
2282:10.1155/2008/685175
2203:. Morgan Kaufmann.
1502:normal distribution
1077:Adaptive mechanisms
887:Parameter selection
404:candidate solutions
236:Genetic Improvement
207:Genetic programming
140:Self-modifying code
95:Gaussian adaptation
4781:Predator satiation
4642:Swarm (simulation)
4637:Swarm intelligence
4612:Agent-based models
4443:Swarming behaviour
4218:Convex programming
4025:Clerc, M. (2004).
3867:. pp. 80β87.
3484:10.1049/cit2.12106
3342:Swarm Intelligence
3311:Van den Bergh, F.
3268:Swarm Intelligence
3192:Swarm Intelligence
3024:Swarm Intelligence
2668:BMC Bioinformatics
2631:Evers, G. (2009).
2319:Clerc, M. (2012).
2296:Zhang, Y. (2015).
2201:Swarm Intelligence
2152:. pp. 69β73.
2052:Fish School Search
2047:Swarm intelligence
1952:
1920:
1900:
1850:
1830:
1798:
1766:
1746:
1714:
1583:
1543:
1523:
1490:
1446:
1417:
1397:
1361:
1322:
981:schools of thought
979:There are several
925:
897:
365:swarm intelligence
311:candidate solution
90:Evolution strategy
70:Cultural algorithm
31:
4814:
4813:
4801:Military swarming
4746:Animal navigation
4665:Collective motion
4652:Collective motion
4519:reverse migration
4453:Swarming motility
4286:
4285:
3774:978-1-905986-10-1
3767:. Luniver Press.
3517:978-1-4799-0454-9
3209:978-3-319-09951-4
3171:(10): 4848-4859.
3041:978-3-319-44426-0
2960:978-0-7803-7282-5
2642:(Master's thesis)
2374:978-1-4244-0708-8
2260:Poli, R. (2008).
2224:Poli, R. (2007).
2210:978-1-55860-595-4
1923:{\displaystyle n}
1769:{\displaystyle L}
1743:
1707:
1686:
1662:
1625:
1520:
1478:
1443:
1420:{\displaystyle i}
1388:
1352:
1300:
1279:
1254:
1247:
1226:
1193:
1160:genetic algorithm
940:meta-optimization
287:
286:
154:Genetic algorithm
115:Memetic algorithm
100:Grammar induction
80:Effective fitness
4839:
4627:Crowd simulation
4604:Swarm algorithms
4575:Insect migration
4480:Animal migration
4472:Animal migration
4465:
4390:Mobbing behavior
4313:
4306:
4299:
4290:
4289:
4200:
4193:
4186:
4177:
4176:
4167:
4158:(5): 9533β9538.
4114:
4113:
4095:
4075:
4069:
4068:
4048:
4042:
4040:Open Archive HAL
4036:
4030:
4023:
4017:
4010:
4004:
3997:
3991:
3990:
3970:
3964:
3963:
3951:
3945:
3944:
3928:
3922:
3921:
3914:
3908:
3901:
3895:
3894:
3860:
3854:
3851:
3820:
3814:
3813:
3785:
3779:
3778:
3760:
3754:
3753:
3736:
3730:
3729:
3701:
3695:
3684:
3678:
3677:
3657:
3651:
3650:
3644:
3635:
3629:
3628:
3626:
3602:
3596:
3585:
3579:
3578:
3558:
3552:
3551:
3545:
3536:
3530:
3529:
3495:
3489:
3488:
3486:
3462:
3456:
3455:
3430:(6): 1362β1381.
3421:
3412:
3403:
3402:
3384:
3375:
3366:
3365:
3339:
3330:
3324:
3323:
3317:
3308:
3302:
3301:
3291:
3263:
3257:
3256:
3220:
3214:
3213:
3187:
3181:
3180:
3156:
3150:
3149:
3147:
3146:
3140:
3129:
3120:
3114:
3107:
3101:
3098:
3092:
3085:
3079:
3072:
3066:
3060:
3054:
3053:
3019:
3013:
3006:
3000:
2993:
2987:
2986:
2980:
2972:
2944:
2928:
2922:
2921:
2919:
2903:
2897:
2896:
2880:
2874:
2873:
2864:. pp. 1β8.
2857:
2851:
2850:
2848:
2820:
2811:
2810:
2790:
2784:
2783:
2773:
2753:
2742:
2741:
2739:
2733:. Archived from
2724:
2713:
2704:
2703:
2693:
2683:
2659:
2653:
2652:
2650:
2649:
2643:
2628:
2619:
2618:
2616:
2598:
2589:
2580:
2579:
2559:
2550:
2549:
2539:
2519:
2506:
2505:
2503:
2492:
2483:
2482:
2480:
2469:
2460:
2451:
2450:
2438:
2432:
2431:
2423:
2414:
2413:
2393:
2387:
2386:
2358:
2347:
2332:
2331:
2325:
2316:
2310:
2309:
2293:
2287:
2286:
2284:
2266:
2257:
2251:
2250:
2248:
2247:
2241:
2235:. Archived from
2230:
2221:
2215:
2214:
2196:
2190:
2189:
2173:
2162:
2161:
2145:
2136:
2135:
2119:
2113:
2112:
2076:
1976:Pareto dominance
1961:
1959:
1958:
1953:
1929:
1927:
1926:
1921:
1909:
1907:
1906:
1901:
1899:
1898:
1889:
1888:
1876:
1875:
1859:
1857:
1856:
1851:
1839:
1837:
1836:
1831:
1807:
1805:
1804:
1799:
1775:
1773:
1772:
1767:
1755:
1753:
1752:
1747:
1745:
1744:
1736:
1723:
1721:
1720:
1715:
1709:
1708:
1700:
1688:
1687:
1679:
1670:
1669:
1664:
1663:
1655:
1633:
1632:
1627:
1626:
1618:
1592:
1590:
1589:
1584:
1582:
1577:
1569:
1564:
1552:
1550:
1549:
1544:
1532:
1530:
1529:
1524:
1522:
1521:
1513:
1499:
1497:
1496:
1491:
1480:
1479:
1471:
1455:
1453:
1452:
1447:
1445:
1444:
1436:
1426:
1424:
1423:
1418:
1406:
1404:
1403:
1398:
1396:
1395:
1390:
1389:
1381:
1370:
1368:
1367:
1362:
1360:
1359:
1354:
1353:
1345:
1331:
1329:
1328:
1323:
1317:
1313:
1312:
1307:
1302:
1301:
1293:
1287:
1286:
1281:
1280:
1272:
1268:
1263:
1255:
1250:
1249:
1248:
1240:
1234:
1233:
1228:
1227:
1219:
1214:
1201:
1200:
1195:
1194:
1186:
934:
932:
931:
928:{\displaystyle }
926:
388:gradient descent
353:social behaviour
279:
272:
265:
251:Parity benchmark
145:Polymorphic code
33:
32:
4847:
4846:
4842:
4841:
4840:
4838:
4837:
4836:
4817:
4816:
4815:
4810:
4729:
4691:
4646:
4598:
4466:
4457:
4322:
4317:
4287:
4282:
4209:
4204:
4123:
4118:
4117:
4093:10.1.1.224.5378
4076:
4072:
4049:
4045:
4037:
4033:
4024:
4020:
4011:
4007:
3998:
3994:
3971:
3967:
3952:
3948:
3929:
3925:
3916:
3915:
3911:
3902:
3898:
3883:
3861:
3857:
3821:
3817:
3786:
3782:
3775:
3761:
3757:
3737:
3733:
3702:
3698:
3685:
3681:
3658:
3654:
3642:
3636:
3632:
3603:
3599:
3586:
3582:
3559:
3555:
3543:
3537:
3533:
3518:
3496:
3492:
3463:
3459:
3419:
3413:
3406:
3382:
3376:
3369:
3337:
3331:
3327:
3315:
3309:
3305:
3264:
3260:
3221:
3217:
3210:
3188:
3184:
3157:
3153:
3144:
3142:
3138:
3127:
3121:
3117:
3108:
3104:
3099:
3095:
3086:
3082:
3073:
3069:
3061:
3057:
3042:
3020:
3016:
3007:
3003:
2994:
2990:
2974:
2973:
2961:
2942:10.1.1.114.7988
2929:
2925:
2917:10.1.1.298.4359
2904:
2900:
2881:
2877:
2858:
2854:
2821:
2814:
2791:
2787:
2771:10.1.1.149.8300
2754:
2745:
2737:
2722:
2714:
2707:
2660:
2656:
2647:
2645:
2641:
2629:
2622:
2596:
2590:
2583:
2560:
2553:
2537:10.1.1.460.6608
2520:
2509:
2501:
2493:
2486:
2478:
2467:
2461:
2454:
2439:
2435:
2424:
2417:
2394:
2390:
2375:
2356:
2348:
2335:
2323:
2317:
2313:
2294:
2290:
2264:
2258:
2254:
2245:
2243:
2239:
2228:
2222:
2218:
2211:
2197:
2193:
2174:
2165:
2146:
2139:
2120:
2116:
2077:
2070:
2065:
2042:Particle filter
2018:
1984:
1968:
1935:
1932:
1931:
1915:
1912:
1911:
1894:
1890:
1884:
1880:
1871:
1867:
1865:
1862:
1861:
1845:
1842:
1841:
1813:
1810:
1809:
1781:
1778:
1777:
1761:
1758:
1757:
1735:
1734:
1732:
1729:
1728:
1699:
1698:
1678:
1677:
1665:
1654:
1653:
1652:
1628:
1617:
1616:
1615:
1613:
1610:
1609:
1599:
1578:
1573:
1565:
1560:
1558:
1555:
1554:
1538:
1535:
1534:
1512:
1511:
1509:
1506:
1505:
1470:
1469:
1461:
1458:
1457:
1435:
1434:
1432:
1429:
1428:
1412:
1409:
1408:
1391:
1380:
1379:
1378:
1376:
1373:
1372:
1355:
1344:
1343:
1342:
1340:
1337:
1336:
1308:
1303:
1292:
1291:
1282:
1271:
1270:
1269:
1264:
1259:
1239:
1238:
1229:
1218:
1217:
1216:
1215:
1213:
1212:
1208:
1196:
1185:
1184:
1183:
1181:
1178:
1177:
1168:
1137:
1135:Simplifications
1124:
1115:
1100:
1091:
1079:
1017:
977:
951:
908:
905:
904:
889:
878:
874:
867:
863:
858:
851:
844:
843:
820:
807:
800:
789:
778:
765:
758:
751:
744:
737:
730:
724:
720:
713:
706:
700:
696:
690: β w
689:
678:
671:
654:each dimension
626:
619:
612:
605:
595:
588:
565:
552:
545:
537:
530:
520:
514:random vector:
484:
478: β β. Let
477:
470:
400:
283:
60:Artificial life
19:
12:
11:
5:
4845:
4835:
4834:
4829:
4827:Metaheuristics
4812:
4811:
4809:
4808:
4803:
4798:
4793:
4788:
4786:Quorum sensing
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4743:
4737:
4735:
4734:Related topics
4731:
4730:
4728:
4727:
4722:
4720:Swarm robotics
4717:
4712:
4707:
4701:
4699:
4697:Swarm robotics
4693:
4692:
4690:
4689:
4684:
4679:
4678:
4677:
4667:
4662:
4656:
4654:
4648:
4647:
4645:
4644:
4639:
4634:
4629:
4624:
4619:
4614:
4608:
4606:
4600:
4599:
4597:
4596:
4591:
4590:
4589:
4588:
4587:
4572:
4571:
4570:
4565:
4555:
4554:
4553:
4548:
4543:
4538:
4531:Fish migration
4528:
4526:Cell migration
4523:
4522:
4521:
4516:
4509:Bird migration
4506:
4505:
4504:
4502:coded wire tag
4499:
4498:
4497:
4487:
4476:
4474:
4468:
4467:
4460:
4458:
4456:
4455:
4450:
4445:
4440:
4439:
4438:
4428:
4427:
4426:
4421:
4411:
4410:
4409:
4399:
4398:
4397:
4395:feeding frenzy
4387:
4382:
4377:
4376:
4375:
4365:
4364:
4363:
4358:
4348:
4343:
4338:
4332:
4330:
4324:
4323:
4316:
4315:
4308:
4301:
4293:
4284:
4283:
4281:
4280:
4275:
4270:
4265:
4263:Metaheuristics
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4214:
4211:
4210:
4203:
4202:
4195:
4188:
4180:
4174:
4173:
4168:
4147:
4141:
4136:
4130:
4122:
4121:External links
4119:
4116:
4115:
4086:(2): 278β300.
4070:
4043:
4031:
4018:
4005:
3992:
3975:Neurocomputing
3965:
3946:
3923:
3909:
3896:
3881:
3855:
3815:
3796:(5): 456β470.
3780:
3773:
3755:
3731:
3712:(4): 919β933.
3696:
3679:
3668:(1): 119β124.
3652:
3630:
3597:
3580:
3569:(1): 183β197.
3553:
3531:
3516:
3490:
3477:(3): 849-862.
3457:
3404:
3393:(6): 832β847.
3367:
3348:(3): 159β198.
3325:
3303:
3258:
3231:(2): 187β216.
3215:
3208:
3182:
3151:
3115:
3102:
3093:
3080:
3067:
3055:
3040:
3014:
3001:
2988:
2959:
2923:
2898:
2875:
2852:
2812:
2785:
2764:(2): 618β628.
2743:
2740:on 2020-02-13.
2705:
2654:
2620:
2581:
2570:(6): 317β325.
2551:
2507:
2484:
2481:on 2003-05-03.
2452:
2433:
2415:
2388:
2373:
2333:
2311:
2288:
2252:
2216:
2209:
2191:
2163:
2137:
2114:
2067:
2066:
2064:
2061:
2060:
2059:
2054:
2049:
2044:
2039:
2034:
2029:
2027:Bees algorithm
2024:
2017:
2014:
2005:
2004:
2001:
1998:
1995:
1983:
1980:
1967:
1964:
1951:
1948:
1945:
1942:
1939:
1919:
1897:
1893:
1887:
1883:
1879:
1874:
1870:
1849:
1829:
1826:
1823:
1820:
1817:
1797:
1794:
1791:
1788:
1785:
1765:
1742:
1739:
1725:
1724:
1713:
1706:
1703:
1697:
1694:
1691:
1685:
1682:
1676:
1673:
1668:
1661:
1658:
1651:
1648:
1645:
1642:
1639:
1636:
1631:
1624:
1621:
1598:
1595:
1581:
1576:
1572:
1568:
1563:
1542:
1519:
1516:
1504:with the mean
1489:
1486:
1483:
1477:
1474:
1468:
1465:
1442:
1439:
1416:
1394:
1387:
1384:
1358:
1351:
1348:
1333:
1332:
1321:
1316:
1311:
1306:
1299:
1296:
1290:
1285:
1278:
1275:
1267:
1262:
1258:
1253:
1246:
1243:
1237:
1232:
1225:
1222:
1211:
1207:
1204:
1199:
1192:
1189:
1167:
1166:Bare Bones PSO
1164:
1156:proven correct
1148:metaheuristics
1136:
1133:
1123:
1120:
1114:
1111:
1098:
1090:
1087:
1078:
1075:
1044:
1043:
1032:
1016:
1013:
976:
975:Inner workings
973:
950:
947:
924:
921:
918:
915:
912:
888:
885:
876:
872:
865:
861:
856:
849:
841:
818:
805:
798:
787:
776:
763:
756:
749:
742:
735:
728:
722:
718:
711:
704:
698:
694:
687:
676:
669:
640:each particle
624:
617:
610:
603:
593:
586:
563:
550:
543:
535:
528:
518:
499:each particle
495:
482:
475:
468:
441:) β€
409:Formally, let
399:
396:
384:differentiable
285:
284:
282:
281:
274:
267:
259:
256:
255:
254:
253:
248:
243:
238:
233:
228:
223:
218:
210:
209:
203:
202:
201:
200:
195:
190:
185:
183:Genetic memory
180:
175:
170:
165:
157:
156:
150:
149:
148:
147:
142:
137:
132:
127:
125:Neuroevolution
122:
117:
112:
107:
102:
97:
92:
87:
82:
77:
72:
67:
62:
57:
49:
48:
42:
41:
27:global minimum
17:
9:
6:
4:
3:
2:
4844:
4833:
4830:
4828:
4825:
4824:
4822:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4738:
4736:
4732:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4702:
4700:
4698:
4694:
4688:
4685:
4683:
4680:
4676:
4673:
4672:
4671:
4668:
4666:
4663:
4661:
4660:Active matter
4658:
4657:
4655:
4653:
4649:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4609:
4607:
4605:
4601:
4595:
4592:
4586:
4583:
4582:
4581:
4578:
4577:
4576:
4573:
4569:
4566:
4564:
4561:
4560:
4559:
4556:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4536:diel vertical
4534:
4533:
4532:
4529:
4527:
4524:
4520:
4517:
4515:
4512:
4511:
4510:
4507:
4503:
4500:
4496:
4493:
4492:
4491:
4488:
4486:
4483:
4482:
4481:
4478:
4477:
4475:
4473:
4469:
4464:
4454:
4451:
4449:
4446:
4444:
4441:
4437:
4434:
4433:
4432:
4429:
4425:
4422:
4420:
4417:
4416:
4415:
4412:
4408:
4405:
4404:
4403:
4400:
4396:
4393:
4392:
4391:
4388:
4386:
4383:
4381:
4378:
4374:
4373:herd behavior
4371:
4370:
4369:
4366:
4362:
4359:
4357:
4354:
4353:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4333:
4331:
4329:
4325:
4321:
4314:
4309:
4307:
4302:
4300:
4295:
4294:
4291:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4215:
4212:
4208:
4201:
4196:
4194:
4189:
4187:
4182:
4181:
4178:
4172:
4169:
4165:
4161:
4157:
4153:
4148:
4145:
4142:
4140:
4137:
4134:
4133:A brief video
4131:
4128:
4125:
4124:
4111:
4107:
4103:
4099:
4094:
4089:
4085:
4081:
4074:
4066:
4062:
4058:
4054:
4047:
4041:
4035:
4028:
4022:
4015:
4009:
4002:
3996:
3988:
3984:
3980:
3976:
3969:
3961:
3957:
3950:
3942:
3938:
3934:
3927:
3919:
3913:
3906:
3900:
3892:
3888:
3884:
3882:0-7803-7914-4
3878:
3874:
3870:
3866:
3859:
3853:
3850:
3848:
3844:
3840:
3836:
3832:
3826:
3819:
3811:
3807:
3803:
3799:
3795:
3791:
3784:
3776:
3770:
3766:
3759:
3751:
3747:
3743:
3735:
3727:
3723:
3719:
3715:
3711:
3707:
3700:
3693:
3689:
3683:
3675:
3671:
3667:
3663:
3656:
3648:
3641:
3634:
3625:
3620:
3616:
3612:
3608:
3601:
3594:
3590:
3584:
3576:
3572:
3568:
3564:
3557:
3549:
3542:
3535:
3527:
3523:
3519:
3513:
3509:
3505:
3501:
3494:
3485:
3480:
3476:
3472:
3468:
3461:
3453:
3449:
3445:
3441:
3437:
3433:
3429:
3425:
3418:
3411:
3409:
3400:
3396:
3392:
3388:
3381:
3374:
3372:
3363:
3359:
3355:
3351:
3347:
3343:
3336:
3329:
3321:
3314:
3307:
3299:
3295:
3290:
3285:
3281:
3277:
3273:
3269:
3262:
3254:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3219:
3211:
3205:
3201:
3197:
3193:
3186:
3178:
3174:
3170:
3166:
3162:
3155:
3141:on 2013-10-23
3137:
3133:
3126:
3119:
3112:
3106:
3097:
3090:
3084:
3078:
3071:
3065:
3059:
3051:
3047:
3043:
3037:
3033:
3029:
3025:
3018:
3011:
3005:
2998:
2992:
2984:
2978:
2970:
2966:
2962:
2956:
2952:
2948:
2943:
2938:
2934:
2927:
2918:
2913:
2909:
2902:
2894:
2890:
2886:
2879:
2871:
2867:
2863:
2856:
2847:
2842:
2838:
2834:
2830:
2826:
2819:
2817:
2808:
2804:
2800:
2796:
2789:
2781:
2777:
2772:
2767:
2763:
2759:
2752:
2750:
2748:
2736:
2732:
2728:
2721:
2720:
2712:
2710:
2701:
2697:
2692:
2687:
2682:
2677:
2673:
2669:
2665:
2658:
2644:on 2011-05-18
2640:
2636:
2635:
2627:
2625:
2615:
2610:
2606:
2602:
2595:
2588:
2586:
2577:
2573:
2569:
2565:
2558:
2556:
2547:
2543:
2538:
2533:
2529:
2525:
2518:
2516:
2514:
2512:
2500:
2499:
2491:
2489:
2477:
2473:
2466:
2459:
2457:
2448:
2444:
2437:
2429:
2422:
2420:
2411:
2407:
2403:
2399:
2392:
2384:
2380:
2376:
2370:
2366:
2362:
2355:
2354:
2346:
2344:
2342:
2340:
2338:
2329:
2322:
2315:
2307:
2303:
2299:
2292:
2283:
2278:
2274:
2270:
2263:
2256:
2242:on 2011-07-16
2238:
2234:
2227:
2220:
2212:
2206:
2202:
2195:
2187:
2183:
2179:
2172:
2170:
2168:
2159:
2155:
2151:
2144:
2142:
2133:
2129:
2125:
2118:
2110:
2106:
2102:
2098:
2094:
2090:
2086:
2082:
2075:
2073:
2068:
2058:
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2019:
2013:
2010:
2002:
1999:
1996:
1993:
1992:
1991:
1988:
1979:
1977:
1973:
1963:
1949:
1946:
1943:
1940:
1937:
1917:
1895:
1891:
1885:
1881:
1877:
1872:
1868:
1847:
1827:
1824:
1821:
1818:
1815:
1795:
1792:
1789:
1786:
1783:
1763:
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1141:Occam's razor
1132:
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1113:Hybridization
1110:
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993:local optimum
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316:
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4705:Ant robotics
4682:Vicsek model
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4144:Applications
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430:
426:
419:real numbers
410:
408:
401:
373:
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319:search-space
298:
294:
288:
109:
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4761:Eusociality
4710:Microbotics
4580:butterflies
4551:sardine run
4485:altitudinal
4407:pack hunter
4059:: 299β308.
3981:: 188β197.
3274:(1): 1β22.
2801:: 148β161.
2404:: 281β295.
2087:(1): 1β54.
1152:empirically
1021:convergence
1015:Convergence
997:convergence
846:The values
361:fish school
307:iteratively
4821:Categories
4675:clustering
4568:philopatry
4546:salmon run
4541:Lessepsian
3833:(6): 781.
3289:2263/62934
3145:2012-04-27
2674:(1): 125.
2648:2010-05-05
2246:2010-05-03
2063:References
1029:converging
449:) for all
433:for which
350:simulating
163:Chromosome
4796:Stigmergy
4776:Mutualism
4436:bait ball
4088:CiteSeerX
3617:: 41β58.
3526:206553432
2977:cite book
2937:CiteSeerX
2912:CiteSeerX
2831:: 70β85.
2766:CiteSeerX
2731:107805461
2532:CiteSeerX
2308:: 931256.
1944:γ
1892:γ
1882:α
1869:α
1848:α
1825:−
1819:∼
1816:α
1793:−
1787:∼
1784:β
1741:→
1705:→
1693:α
1684:→
1675:β
1660:→
1647:β
1644:−
1635:←
1623:→
1571:…
1541:σ
1518:→
1485:σ
1476:→
1441:→
1386:→
1350:→
1298:→
1289:−
1277:→
1245:→
1224:→
1191:→
1063:empirical
614:|, |
398:Algorithm
374:PSO is a
315:particles
303:optimizes
193:Selection
173:Crossover
4725:Symbrion
4687:BIO-LGCA
4490:tracking
4419:ant mill
4361:sort sol
4356:flocking
4320:Swarming
4110:17984726
3891:37185749
3810:22382958
3726:27974467
3452:11191625
3444:19362911
3253:25471827
3245:24738856
3050:37588745
2969:14364974
2700:16529661
2607:: 1β10.
2275:: 1β10.
2101:26953883
2016:See also
1910:, where
1089:Variants
489:and let
423:gradient
380:gradient
342:Eberhart
331:velocity
327:position
178:Mutation
38:a series
36:Part of
4585:monarch
4514:flyways
4495:history
4346:Droving
4146:of PSO.
3847:2864886
3362:2261683
3298:9778346
2691:1464136
2383:6217309
2109:8783143
1500:is the
875:, and Ο
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779:) <
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338:Kennedy
246:Eurisko
4558:Homing
4380:Locust
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415:vector
241:Schema
40:on the
4622:Boids
4563:natal
4351:Flock
4106:S2CID
3887:S2CID
3843:S2CID
3806:S2CID
3722:S2CID
3643:(PDF)
3544:(PDF)
3522:S2CID
3448:S2CID
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3358:S2CID
3338:(PDF)
3316:(PDF)
3294:S2CID
3249:S2CID
3139:(PDF)
3128:(PDF)
3062:SPSO
3046:S2CID
2965:S2CID
2727:S2CID
2723:(PDF)
2597:(PDF)
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2379:S2CID
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2105:S2CID
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1001:below
991:to a
881:below
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630:while
357:flock
4402:Pack
4368:Herd
3877:ISBN
3769:ISBN
3512:ISBN
3440:PMID
3241:PMID
3204:ISBN
3036:ISBN
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2955:ISBN
2696:PMID
2605:2008
2369:ISBN
2306:2015
2273:2008
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2097:PMID
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