Particle swarm optimization

In computer science, particle swarm optimization (PSO) is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, here dubbed particles, and moving these particles around in the search-space according to simple mathematical formulae over the particle's position and velocity. 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.

PSO is originally attributed to Kennedy, Eberhart and Shi[1][2] and was first intended for simulating social behaviour,[3] as a stylized representation of the movement of organisms in a bird flock or fish school. The algorithm was simplified and it was observed to be performing optimization. The book by Kennedy and Eberhart[4] describes many philosophical aspects of PSO and swarm intelligence. An extensive survey of PSO applications is made by Poli.[5][6] Recently, a comprehensive review on theoretical and experimental works on PSO has been published by Bonyadi and Michalewicz.[7]

PSO is a metaheuristic as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. However, metaheuristics such as PSO do not guarantee an optimal solution is ever found. More specifically, PSO does not use the gradient of the problem being optimized, which means PSO does not require that the optimization problem be differentiable as is required by classic optimization methods such as gradient descent and quasi-newton methods.

Algorithm

A basic variant of the PSO algorithm works by having a population (called a swarm) of candidate solutions (called particles). These particles are moved around in the search-space according to a few simple formulae.[8] 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.

Formally, let f: ℝn → ℝ be the cost function which must be minimized. The function takes a candidate solution as argument in the form of a vector of real numbers and produces a real number as output which indicates the objective function value of the given candidate solution. The gradient of f is not known. The goal is to find a solution a for which f(a) ≤ f(b) for all b in the search-space, which would mean a is the global minimum. Maximization can be performed by considering the function h = -f instead.

Let S be the number of particles in the swarm, each having a position xi ∈ ℝn in the search-space and a velocity vi ∈ ℝn. Let pi be the best known position of particle i and let g be the best known position of the entire swarm. A basic PSO algorithm is then:[9]

for each particle i = 1, ..., S do
   Initialize the particle's position with a uniformly distributed random vector: xi ~ U(blo, bup)
   Initialize the particle's best known position to its initial position: pi ← xi
   if f(pi) < f(g) then
       update the swarm's best known  position: g ← pi
   Initialize the particle's velocity: vi ~ U(-|bup-blo|, |bup-blo|)
while a termination criterion is not met do:
   for each particle i = 1, ..., S do
      for each dimension d = 1, ..., n do
         Pick random numbers: rp, rg ~ U(0,1)
         Update the particle's velocity: vi,d ← ω vi,d + φp rp (pi,d-xi,d) + φg rg (gd-xi,d)
         Update the particle's position: xi ← xi + vi
         if f(xi) < f(pi) then
            Update the particle's best known position: pi ← xi
            if f(pi) < f(g) then
               Update the swarm's best known position: g ← pi

The values blo and bup are respectively the lower and upper boundaries of the search-space. The termination criterion can be number of iterations performed, or a solution with adequate objective function value is found.[10] The parameters ω, φp, and φg are selected by the practitioner and control the behaviour and efficacy of the PSO method, see below.

Parameter selection

Performance landscape showing how a simple PSO variant performs in aggregate on several benchmark problems when varying two PSO parameters.

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.[11][12][13][14][15][16][17][18][19]

The PSO parameters can also be tuned by using another overlaying optimizer, a concept known as meta-optimization.[20][21][22] Parameters have also been tuned for various optimization scenarios.[23]

Neighborhoods and topologies

The topology of the swarm defines the subset of particles which each particle can exchange information.[24] The basic version of the algorithm uses the global topology as the swarm communication structure.[10] This topology allows all particles to communicate with all the other particles, thus the whole swarm share the same best position g from a single particle. However, this approach might lead the swarm to be trapped into a local minimum,[25] 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.[10] This subset can be a geometrical one[26] – for example "the m nearest particles" – or, more often, a social one, i.e. a set of particles that is not depending on any distance. In such a case, the PSO variant is said to be local best (vs global best for the basic PSO).

A commonly used swarm topology is the ring, in which each particle has just two neighbours, but there are many others.[10] The topology is not necessarily static. In fact, since the topology is related to the diversity of communication of the particles,[27] some efforts have been done to create adaptive topologies (SPSO,[28] stochastic star,[29] TRIBES,[30] Cyber Swarm,[31] and C-PSO[32]).

Inner workings

There are several schools of thought as to why and how the PSO algorithm can perform optimization.

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 been prevalent since the inception of PSO.[2][3][12][16] 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 premature convergence to a local optimum yet still ensure a good rate of convergence to the optimum. This belief is the precursor of many PSO variants, see below.

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 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 below.

Convergence

In relation to PSO the word convergence typically refers to two different definitions:

Convergence of the sequence of solutions has been investigated for PSO.[15][16][17] 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 somewhere). However, the analyses were criticized by Pedersen[22] 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 p and the swarm's best known position g, remain constant throughout the optimization process. However, it was shown[33] that these simplifications do not affect the boundaries found by these studies for parameter where the swarm is convergent.

Convergence to a local optimum has been analyzed for PSO in[34] and.[35] It has been proven that PSO need some modification to guarantee to find a local optimum.

This means that determining convergence capabilities of different PSO algorithms and parameters therefore still depends on empirical 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 p and g, 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.[36] However, such studies do not provide theoretical evidence to actually prove their claims.

Biases

As the basic PSO works dimension by dimension, the solution point is easier found when it lies on an axis of the search space, on a diagonal, and even easier if it is right on the centre.[37][38]

One approach is to modify the algorithm so that it is not any more sensitive to the system of coordinates.[39][40][41][42] Note that some of these methods have a higher computational complexity (are in O(n^2) where n is the number of dimensions) that make the algorithm very slow for large scale optimization.[35]

The only currently existing PSO variant that is not sensitive to the rotation of the coordinates while is locally convergent has been proposed at 2014.[35] The method has shown a very good performance on many benchmark problems while its rotation invariance and local convergence have been mathematically proven.

Variants

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 pi and g after the entire swarm has been updated, etc. Some of these choices and their possible performance impact have been discussed in the literature.[14]

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 provides a valuable point of comparison which can be used throughout the field of research to better test new advances."[10] The latest is Standard PSO 2011 (SPSO-2011).[43]

Hybridization

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,[44][45][46] e.g., combined PSO with biogeography-based optimization,[47] and the incorporation of an effective learning method.[36]

Alleviate premature

Another research trend is to try and alleviate premature convergence (that is, optimization stagnation), e.g. by reversing or perturbing the movement of the PSO particles,[19][48][49][50] another approach to deal with premature convergence is the use of multiple swarms[51] (multi-swarm optimization). The multi-swarm approach can also be used to implement multi-objective optimization.[52] Finally, there are developments in adapting the behavioural parameters of PSO during optimization.[53]

Simplifications

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 Occam's razor. Simplifying PSO was originally suggested by Kennedy[3] and has been studied more extensively,[18][21][22][54] where it appeared that optimization performance was improved, and the parameters were easier to tune and they performed more consistently across different optimization problems.

Another argument in favour of simplifying PSO is that metaheuristics can only have their efficacy demonstrated empirically by doing computational experiments on a finite number of optimization problems. This means a metaheuristic such as PSO cannot be proven correct and this increases the risk of making errors in its description and implementation. A good example of this[55] presented a promising variant of a genetic algorithm (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.[56]

Initialization of velocities may require extra inputs. The Bare Bones PSO variant[57] has been proposed in 2003 by James Kennedy, and does not need to use velocity at all.

Another simpler variant is the accelerated particle swarm optimization (APSO),[58] 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.[59]

Multi-objective optimization

PSO has also been applied to multi-objective problems,[60][61] in which the objective function comparison takes pareto dominance into account when moving the PSO particles and non-dominated solutions are stored so as to approximate the pareto front.

Binary, discrete, and combinatorial

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 more sophisticated.[62]

However, it can be noted that the equations of movement make use of operators that perform four actions:

Usually a position and a velocity are represented by n 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.[63][64][65][66] One approach is to redefine the operators based on sets.[67]

See also

References

  1. ↑ Kennedy, J.; Eberhart, R. (1995). "Particle Swarm Optimization". Proceedings of IEEE International Conference on Neural Networks. IV. pp. 1942–1948. doi:10.1109/ICNN.1995.488968.
  2. 1 2 Shi, Y.; Eberhart, R.C. (1998). "A modified particle swarm optimizer". Proceedings of IEEE International Conference on Evolutionary Computation. pp. 69–73.
  3. 1 2 3 Kennedy, J. (1997). "The particle swarm: social adaptation of knowledge". Proceedings of IEEE International Conference on Evolutionary Computation. pp. 303–308.
  4. ↑ Kennedy, J.; Eberhart, R.C. (2001). Swarm Intelligence. Morgan Kaufmann. ISBN 1-55860-595-9.
  5. ↑ Poli, R. (2007). "An analysis of publications on particle swarm optimisation applications" (PDF). Technical Report CSM-469. Department of Computer Science, University of Essex, UK.
  6. ↑ Poli, R. (2008). "Analysis of the publications on the applications of particle swarm optimisation" (PDF). Journal of Artificial Evolution and Applications. 2008: 1–10. doi:10.1155/2008/685175.
  7. ↑ Bonyadi, M. R.; Michalewicz, Z. (2016). "Particle swarm optimization for single objective continuous space problems: a review". Evolutionary Computation. in press.
  8. ↑ Zhang, Y. (2015). "A Comprehensive Survey on Particle Swarm Optimization Algorithm and Its Applications". Mathematical Problems in Engineering. 2015: 931256.
  9. ↑ Clerc, M. (2012). "Standard Particle Swarm Optimisation" (PDF). HAL open access archive.
  10. 1 2 3 4 5 Bratton, Daniel; Kennedy, James (2007). "Defining a Standard for Particle Swarm Optimization" (PDF). Proceedings of the 2007 IEEE Swarm Intelligence Symposium (SIS 2007).
  11. ↑ Taherkhani, M.; Safabakhsh, R. (2016). "A novel stability-based adaptive inertia weight for particle swarm optimization". Applied Soft Computing. 38: 281–295. doi:10.1016/j.asoc.2015.10.004.
  12. 1 2 Shi, Y.; Eberhart, R.C. (1998). "Parameter selection in particle swarm optimization". Proceedings of Evolutionary Programming VII (EP98). pp. 591–600.
  13. ↑ Eberhart, R.C.; Shi, Y. (2000). "Comparing inertia weights and constriction factors in particle swarm optimization". Proceedings of the Congress on Evolutionary Computation. 1. pp. 84–88.
  14. 1 2 Carlisle, A.; Dozier, G. (2001). "An Off-The-Shelf PSO" (PDF). Proceedings of the Particle Swarm Optimization Workshop. pp. 1–6.
  15. 1 2 van den Bergh, F. (2001). An Analysis of Particle Swarm Optimizers (PhD thesis). University of Pretoria, Faculty of Natural and Agricultural Science.
  16. 1 2 3 Clerc, M.; Kennedy, J. (2002). "The particle swarm - explosion, stability, and convergence in a multidimensional complex space". IEEE Transactions on Evolutionary Computation. 6 (1): 58–73. doi:10.1109/4235.985692.
  17. 1 2 Trelea, I.C. (2003). "The Particle Swarm Optimization Algorithm: convergence analysis and parameter selection". Information Processing Letters. 85 (6): 317–325. doi:10.1016/S0020-0190(02)00447-7.
  18. 1 2 Bratton, D.; Blackwell, T. (2008). "A Simplified Recombinant PSO". Journal of Artificial Evolution and Applications.
  19. 1 2 Evers, G. (2009). An Automatic Regrouping Mechanism to Deal with Stagnation in Particle Swarm Optimization (Master's thesis). The University of Texas - Pan American, Department of Electrical Engineering.
  20. ↑ Meissner, M.; Schmuker, M.; Schneider, G. (2006). "Optimized Particle Swarm Optimization (OPSO) and its application to artificial neural network training". BMC Bioinformatics. 7 (1): 125. doi:10.1186/1471-2105-7-125. PMC 1464136Freely accessible. PMID 16529661.
  21. 1 2 Pedersen, M.E.H. (2010). Tuning & Simplifying Heuristical Optimization (PhD thesis). University of Southampton, School of Engineering Sciences, Computational Engineering and Design Group.
  22. 1 2 3 Pedersen, M.E.H.; Chipperfield, A.J. (2010). "Simplifying particle swarm optimization" (PDF). Applied Soft Computing. 10 (2): 618–628. doi:10.1016/j.asoc.2009.08.029.
  23. ↑ Pedersen, M.E.H. (2010). "Good parameters for particle swarm optimization" (PDF). Technical Report HL1001. Hvass Laboratories.
  24. ↑ Kennedy, J.; Mendes, R. (2002). "Population structure and particle swarm performance". Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on. doi:10.1109/CEC.2002.1004493.
  25. ↑ Mendes, R. (2004). Population Topologies and Their Influence in Particle Swarm Performance (PhD thesis). Universidade do Minho.
  26. ↑ Suganthan, Ponnuthurai N. "Particle swarm optimiser with neighbourhood operator." Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on. Vol. 3. IEEE, 1999.
  27. ↑ Oliveira, M.; Pinheiro, D.; Andrade, B.; Bastos-Filho, C.; Menezes, R. (2016). "Communication Diversity in Particle Swarm Optimizers". International Conference on Swarm Intelligence. doi:10.1007/978-3-319-44427-7_7.
  28. ↑ SPSO Particle Swarm Central
  29. ↑ Miranda, V., Keko, H. and Duque, Á. J. (2008). Stochastic Star Communication Topology in Evolutionary Particle Swarms (EPSO). International Journal of Computational Intelligence Research (IJCIR), Volume 4, Number 2, pp. 105-116
  30. ↑ Clerc, M. (2006). Particle Swarm Optimization. ISTE (International Scientific and Technical Encyclopedia), 2006
  31. ↑ Yin, P., Glover, F., Laguna, M., & Zhu, J. (2011). A Complementary Cyber Swarm Algorithm. International Journal of Swarm Intelligence Research (IJSIR), 2(2), 22-41
  32. ↑ Elshamy, W.; Rashad, H.; Bahgat, A. (2007). "Clubs-based Particle Swarm Optimization" (PDF). IEEE Swarm Intelligence Symposium 2007 (SIS2007). Honolulu, HI. pp. 289–296.
  33. ↑ Cleghorn, Christopher W (2014). "Particle Swarm Convergence: Standardized Analysis and Topological Influence". Swarm Intelligence Conference.
  34. ↑ Van den Bergh, F. "A convergence proof for the particle swarm optimiser". Fundamenta Informaticae.
  35. 1 2 3 Bonyadi, Mohammad reza.; Michalewicz, Z. (2014). "A locally convergent rotationally invariant particle swarm optimization algorithm". Swarm intelligence. 8 (3): 159–198. doi:10.1007/s11721-014-0095-1.
  36. 1 2 Zhan, Z-H.; Zhang, J.; Li, Y; Shi, Y-H. (2011). "Orthogonal Learning Particle Swarm Optimization" (PDF). IEEE Transactions on Evolutionary Computation. 15 (6): 832–847. doi:10.1109/TEVC.2010.2052054.
  37. ↑ Monson, C. K. & Seppi, K. D. (2005). Exposing Origin-Seeking Bias in PSO GECCO'05, pp. 241-248
  38. ↑ Spears, W. M., Green, D. T. & Spears, D. F. (2010). Biases in Particle Swarm Optimization. International Journal of Swarm Intelligence Research, Vol. 1(2), pp. 34-57
  39. ↑ Wilke, D. N., Kok, S. & Groenwold, A. A. (2007). Comparison of linear and classical velocity update rules in particle swarm optimization: notes on scale and frame invariance. International Journal for Numerical Methods in Engineering, John Wiley & Sons, Ltd., 70, pp. 985-1008
  40. ↑ SPSO 2011, Particle Swarm Central
  41. ↑ Bonyadi, Mohammad reza; Michalewicz, Z. (2014). "SPSO 2011 analysis of stability; local convergence; and rotation sensitivity.". GECCO2014 (the best paper award in the track ACSI): 9–16.
  42. ↑ Bonyadi, Mohammad reza.; Michalewicz, Z. (2014). "An analysis of the velocity updating rule of the particle swarm optimization algorithm". Journal of Heuristics. 20 (4): 417–452. doi:10.1007/s10732-014-9245-2.
  43. ↑ Zambrano-Bigiarini, M.; Clerc, M.; Rojas, R. (2013). "Standard Particle Swarm Optimisation 2011 at CEC-2013: A baseline for future PSO improvements". Evolutionary Computation (CEC), 2013 IEEE Congress on.
  44. ↑ Lovbjerg, M.; Krink, T. (2002). "The LifeCycle Model: combining particle swarm optimisation, genetic algorithms and hillclimbers". Proceedings of Parallel Problem Solving from Nature VII (PPSN). pp. 621–630.
  45. ↑ Niknam, T.; Amiri, B. (2010). "An efficient hybrid approach based on PSO, ACO and k-means for cluster analysis". Applied Soft Computing. 10 (1): 183–197. doi:10.1016/j.asoc.2009.07.001.
  46. ↑ Zhang, Wen-Jun; Xie, Xiao-Feng (2003). DEPSO: hybrid particle swarm with differential evolution operator. IEEE International Conference on Systems, Man, and Cybernetics (SMCC), Washington, DC, USA: 3816-3821.
  47. ↑ Zhang, Y.; Wang, S. (2015). "Pathological Brain Detection in Magnetic Resonance Imaging Scanning by Wavelet Entropy and Hybridization of Biogeography-based Optimization and Particle Swarm Optimization". Progress in Electromagnetics Research – Pier. 152: 41–58.
  48. ↑ Lovbjerg, M.; Krink, T. (2002). "Extending Particle Swarm Optimisers with Self-Organized Criticality". Proceedings of the Fourth Congress on Evolutionary Computation (CEC). 2. pp. 1588–1593.
  49. ↑ Xinchao, Z. (2010). "A perturbed particle swarm algorithm for numerical optimization". Applied Soft Computing. 10 (1): 119–124. doi:10.1016/j.asoc.2009.06.010.
  50. ↑ Xie, Xiao-Feng; Zhang, Wen-Jun; Yang, Zhi-Lian (2002). A dissipative particle swarm optimization. Congress on Evolutionary Computation (CEC), Honolulu, HI, USA: 1456-1461.
  51. ↑ Cheung, N. J., Ding, X.-M., & Shen, H.-B. (2013). OptiFel: A Convergent Heterogeneous Particle Sarm Optimization Algorithm for Takagi-Sugeno Fuzzy Modeling, IEEE Transactions on Fuzzy Systems, doi:10.1109/TFUZZ.2013.2278972
  52. ↑ 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". Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics. Lecture Notes in Computer Science. 7264. pp. 74–85.
  53. ↑ Zhan, Z-H.; Zhang, J.; Li, Y; Chung, H.S-H. (2009). "Adaptive Particle Swarm Optimization" (PDF). IEEE Transactions on Systems, Man, and Cybernetics. 39 (6): 1362–1381. doi:10.1109/TSMCB.2009.2015956.
  54. ↑ Yang, X.S. (2008). Nature-Inspired Metaheuristic Algorithms. Luniver Press. ISBN 978-1-905986-10-1.
  55. ↑ Tu, Z.; Lu, Y. (2004). "A robust stochastic genetic algorithm (StGA) for global numerical optimization". IEEE Transactions on Evolutionary Computation. 8 (5): 456–470. doi:10.1109/TEVC.2004.831258.
  56. ↑ Tu, Z.; Lu, Y. (2008). "Corrections to "A Robust Stochastic Genetic Algorithm (StGA) for Global Numerical Optimization". IEEE Transactions on Evolutionary Computation. 12 (6): 781–781. doi:10.1109/TEVC.2008.926734.
  57. ↑ Kennedy, James (2003). "Bare Bones Particle Swarms". Proceedings of the 2003 IEEE Swarm Intelligence Symposium.
  58. ↑ X. S. Yang, S. Deb and S. Fong, Accelerated particle swarm optimization and support vector machine for business optimization and applications, NDT 2011, Springer CCIS 136, pp. 53-66 (2011).
  59. ↑ http://www.mathworks.com/matlabcentral/fileexchange/?term=APSO
  60. ↑ Parsopoulos, K.; Vrahatis, M. (2002). "Particle swarm optimization method in multiobjective problems". Proceedings of the ACM Symposium on Applied Computing (SAC). pp. 603–607.
  61. ↑ Coello Coello, C.; Salazar Lechuga, M. (2002). "MOPSO: A Proposal for Multiple Objective Particle Swarm Optimization". Congress on Evolutionary Computation (CEC'2002). pp. 1051–1056.
  62. ↑ Roy, R., Dehuri, S., & Cho, S. B. (2012). A Novel Particle Swarm Optimization Algorithm for Multi-Objective Combinatorial Optimization Problem. 'International Journal of Applied Metaheuristic Computing (IJAMC)', 2(4), 41-57
  63. ↑ Kennedy, J. & Eberhart, R. C. (1997). A discrete binary version of the particle swarm algorithm, Conference on Systems, Man, and Cybernetics, Piscataway, NJ: IEEE Service Center, pp. 4104-4109
  64. ↑ Clerc, M. (2004). Discrete Particle Swarm Optimization, illustrated by the Traveling Salesman Problem, New Optimization Techniques in Engineering, Springer, pp. 219-239
  65. ↑ Clerc, M. (2005). Binary Particle Swarm Optimisers: toolbox, derivations, and mathematical insights, Open Archive HAL
  66. ↑ Jarboui, B., Damak, N., Siarry, P., and Rebai, A.R. (2008). A combinatorial particle swarm optimization for solving multi-mode resource-constrained project scheduling problems. In Proceedings of Applied Mathematics and Computation, pp. 299-308.
  67. ↑ Chen, Wei-neng; Zhang, Jun (2010). "A novel set-based particle swarm optimization method for discrete optimization problem". IEEE Transactions on Evolutionary Computation. 14 (2): 278–300. doi:10.1109/tevc.2009.2030331.

External links

This article is issued from Wikipedia - version of the 9/23/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.