zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Particle swarm optimization with fractional-order velocity. (English) Zbl 1204.90119
Summary: This paper proposes a novel method for controlling the convergence rate of a particle swarm optimization algorithm using fractional calculus (FC) concepts. The optimization is tested for several well-known functions and the relationship between the fractional order velocity and the convergence of the algorithm is observed. The FC demonstrates a potential for interpreting evolution of the algorithm and to control its convergence.
MSC:
90C59Approximation methods and heuristics
26A33Fractional derivatives and integrals (real functions)
References:
[1]Banks, A., Vincent, J., Anyakoha, C.: A review of particle swarm optimization. ii: Hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications. Nat. Comput. 7(1), 109–124 (2008) · Zbl 1148.68375 · doi:10.1007/s11047-007-9050-z
[2]Gement, A.: On fractional differentials. Proc. Philos. Mag. 25, 540–549 (1938)
[3]Oustaloup, A.: La Commande CRONE: Commande Robuste d’Ordre Non Intier. Hermes, Paris (1991)
[4]Méhauté, A.L.: Fractal Geometries: Theory and Applications. Penton Press, Cleveland (1991)
[5]Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
[6]Tenreiro Machado, J.A.: Analysis and design of fractional-order digital control systems. J. Syst. Anal.-Model. Simul. 27, 107–122 (1997)
[7]Tenreiro Machado, J.A.: System modeling and control through fractional-order algorithms. FCAA–J. Fractional Calc. Appl. Anal. 4, 47–66 (2001)
[8]Vinagre, B.M., Petras, I., Podlubny, I., Chen, Y.Q.: Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn. 1–4(29), 269–279 (2002) · Zbl 1031.93110 · doi:10.1023/A:1016504620249
[9]Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behaviour of real materials. ASME J. Appl. Mech. 51, 294–298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615
[10]Westerlund, S.: Dead Matter Has Memory! Causal Consulting. Kalmar, Sweden (2002)
[11]Herzallah, M.A.E., Baleanu, D.: Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn. 58(1–2), 385–391 (2009) · Zbl 1183.26006 · doi:10.1007/s11071-009-9486-z
[12]Rabei, E.M., Altarazi, I.M.A., Muslih, S.I., Baleanu, D.: Fractional WKB approximation. Nonlinear Dyn. 57(1–2), 171–175 (2009) · Zbl 1176.70014 · doi:10.1007/s11071-008-9430-7
[13]Tarasov, V.E., Zaslavsky, G.M.: Fokker-Planck equation with fractional coordinate derivatives. Physica A, Stat. Mech. Appl. 387(26), 6505–6512 (2008) · doi:10.1016/j.physa.2008.08.033
[14]Magin, R., Feng, X., Baleanu, D.: Solving the fractional order Bloch equation. Concepts Magn. Reson. 34A(1), 16–23 (2009) · doi:10.1002/cmr.a.20129
[15]Solteiro Pires, E.J., de Moura Oliveira, P.B., Tenreiro Machado, J.A., Jesus, I.S.: Fractional order dynamics in a particle swarm optimization algorithm. In: Seventh International Conference on Intelligent Systems Design and Applications, ISDA 2007, Washington, DC, USA, pp. 703–710. IEEE Computer Society, Los Alamitos (2007)
[16]Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, vol. 4, pp. 1942–1948. IEEE Service Center, Piscataway (1995)
[17]Shi, Y., Eberhart, R.: A modified particle swarm optimizer. In: Evolutionary Computation Proceedings, 1998. IEEE World Congress on Computational Intelligence, The 1998 IEEE International Conference on, Anchorage, Alaska, pp. 69–73 (1998)
[18]Løvbjerg, M., Rasmussen, T.K., Krink, T.: Hybrid particle swarm optimiser with breeding and subpopulations. In: Spector, L., Goodman, E.D., Wu, A., Langdon, W., Voigt, H.M., Gen, M., Sen, S., Dorigo, M., Pezeshk, S., Garzon, M.H., Burke, E. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2001), San Francisco, California, USA (7–11 July), pp. 469–476. Morgan Kaufmann, San Mateo (2001)
[19]Solteiro Pires, E.J., Tenreiro Machado, J.A., de Moura Oliveira, P.B., Reis, C.: Fractional dynamics in particle swarm optimization. In: ISIC. IEEE International Conference on Systems, Man and Cybernetics, Montreal, Que. (7–10 Oct. 2007), pp. 1958–1962
[20]Reis, C., Machado, J., Galhano, A., Cunha, J.: Circuit synthesis using particle swarm optimization. In: IEEE International Conference on Computational Cybernetics (ICCC 2006) (Aug. 2006), pp. 1–6
[21]Eberhart, R., Simpson, P., Dobbins, R.: Computational Intelligence PC Tools. Academic Press, San Diego (1996)
[22]den Bergh, F.V., Engelbrecht, A.P.: A study of particle swarm optimization particle trajectories. Inf. Sci. 176(8), 937–971 (2006) · Zbl 1093.68105 · doi:10.1016/j.ins.2005.02.003
[23]Eberhart, R., Shi, Y.: Comparing inertia weights and constriction factors in particle swarm optimization. In: Proceedings of the 2000 Congress on Evolutionary Computation, Washington, DC, vol. 1, pp. 84–88 (2000)
[24]Clerc, M., Kennedy, J.: The particle swarm–explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002) · Zbl 05451976 · doi:10.1109/4235.985692