No Access

COMPUTING TWO LINCHPINS OF TOPOLOGICAL DEGREE BY A NOVEL DIFFERENTIAL EVOLUTION ALGORITHM

FEI GAO

Department of Mathematics, Wuhan University of Technology, Wuhan, Hubei 430070, P. R. China

Search for more papers by this author

and

HENG-QING TONG

Department of Mathematics, Wuhan University of Technology, Wuhan, Hubei 430070, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S1469026805001647Cited by:8 (Source: Crossref)

Abstract

How to detect the topological degree (TD) of a function is of vital importance in investigating the existence and the number of zero values in the function, which is a topic of major significance in the theory of nonlinear scientific fields. Usually a sufficient refinement of the boundary of the polyhedron decided by Boult and Sikorski algorithm (BS) is needed as prerequisite when the well known method of Stenger and Kearfott is chosen for computing TD. However two linchpins are indispensable to BS, the parameter δ on the boundary of the polyhedron and an estimation of the Lipschitz constant K of the function, whose computations are analytically difficult. In this paper, through an appropriate scheme that transforms the problems of computing δ and K into searching optimums of two non-differentiable functions, a novel differential evolution algorithm (DE) combined with established techniques is proposed as an alternative method to computing δ and K. Firstly it uses uniform design method to generate the initial population in feasible field so as to have the property of large scale convergence, without better approximation of the unknown parameter as iterative initial point. Secondly, it restrains the normal DE's local convergence limitation virtually through deflection and stretching of objective function. The main advantages of the put algorithm are its simplicity and its ability to work by using function values solely. Finally, details of applying the proposed method into computing δ and K are given, and experimental results on two benchmark problems in contrast to the results reported have demonstrated the promising performance of the proposed algorithm in different scenarios.

Keywords:

Remember to check out the Most Cited Articles!
Check out these titles in artificial intelligence!

References

T. Boult and K. Sikorski, SIAM J. Sci. Statist. Comput. 10, 686 (1989). Crossref, Google Scholar
F. Stenger, Numer. Math. 25, 23 (1975). Crossref, Google Scholar
R. Storn and K. Price , J. Global Optimization 11 , 341 . Crossref, Google Scholar
K. Price, New Ideas in Optimization, eds. D. Corne, M. Dorigo and F. Glover (McGraw-Hill, London, UK, 1999) pp. 79–108. Google Scholar
R. Storn, Differential evolution design of an IIR2filter, IEEE Int. Conf. Evol. Comput. (1996) pp. 268–273. Google Scholar
X. F. Xie, W. J. Zhang and D. C. Bi, Congress on Evolutionary Computation (CEC) (Oregon, USA, 2004) pp. 2012–2016. Google Scholar
X. F. Xieet al., Contr. Decision 19(1), 49 (2004). Google Scholar
Wikipedia, Category: Optimization http://en.wikipedia.org/wiki/Category: Optimization (2005) . Google Scholar
Y. X. Yuan and W. Y. Sun , Optimization Theory and Method ( Science Press , Peiking , 1997 ) . Google Scholar
D. Cvijovi and J. Klinowski, Science 267, 664 (1995). Crossref, Google Scholar
A. Dekkers and E. Aarts, Math. Program. 50, 367 (1991). Crossref, Google Scholar
H. G. Beyer and H. P. Schwefel, Nat. Comput. 1, 35 (2002). Google Scholar
D. C. Wunsch II and S. Mulder, Evolutionary algorithms, Markov decision processes, adaptive critic designs, and clustering: Commonalities, hybridization and performance, Intelligent Sensing and Information Processing, Proceedings of International Conference on 2004 (2004) pp. 477–482. Google Scholar
J. Kennedy and R. C. Eberhart, Particle swarm optimization, IEEE Int. Conf. Neural Networks (1995) pp. 1942–1948. Google Scholar
F. Gao, Complex Syst. Complexity Sci. 1(4), 79 (2004). Google Scholar
R. Storn, Differential evolution for continuous function optimization, http://www.icsi. berkeley.edu/~storn/code.html (2005) . Google Scholar
C. X. Ma, Uniform design based on centered l.2 discrepancy Un(ns), http://www. math.hkbu.edu.hk/UniformDesign (1999.9) . Google Scholar
K. Parsopoulos and M. Vrahatis , Computing periodic orbits of nonlinear mappings through particle swarm optimization , Proc. 4th GRACM Congr. Comput. Mech. ( 2002 ) . Google Scholar
N. G. Lloyd , Degree Theory ( Cambridge University Press , Cambridge , 1978 ) . Google Scholar
M. N. Vrahatis, Proc. Amer. Math. Soc. 107, 701 (1989). Google Scholar
M. N. Vrahatis, J. Comput. Phys. 119, 105 (1995). Crossref, Google Scholar
M. N. Vrahatis, T. C. Bountis and M. Kollmann, Int. J. Bifurcat. Chaos 6, 1425 (1996). Link, Google Scholar
M. N. Vrahatis, H. Isliker and T. C. Bountis, Int. J. Bifurcat. Chaos 7, 2707 (1997). Link, Google Scholar
K. E. Parsopoulos and M. N. Vrahatis, Investigating the existence of function roots using particle swarm optimization, Proc. IEEE 2003 Congr. Evol. Comput. (2003) pp. 1448–1455. Google Scholar
Wolfram Research, Inc., Sign function, http://functions.wolfram.com/Complex Components/Sign/02/ (2005) . Google Scholar
C. Polymilis, G. Servizi, Ch. Skokos et al. Locating periodic orbits by topological degree theory, http://arxiv.org/abs/nlin/0211044 (2002) . Google Scholar
E. Picard , Traité d' analyse , 3rd edn. ( Gauthier-Villars , Paris , 1922 ) . Google Scholar
Wolfram Research, Inc., Gamma function, http://mathworld.wolfram.com/Gamma Function.html (2005) . Google Scholar
E. Picard, J. Math. Pure Appl. (4e s'erie) 8, 5 (1892). Google Scholar
E. Picard , Trait'e d'analyse ( Gauthier-Villars , Paris , 1922 ) . Google Scholar
M. N. Vrahatiset al., Numer. Funct. Anal. Optimization 18, 227 (1997). Crossref, Google Scholar
B. Mourrain, M. N. Vrahatis and J. C. Yakoubsohn, J. Complexity 18, 612 (2002). Crossref, Google Scholar
K. A. Sikorski , Optimal Solution of Nonlinear Equations ( Oxford University Press , 2001 ) . Crossref, Google Scholar
Wolfram Research, Inc., Lipschitz Function, http://mathworld.wolfram.com/LipschitzFunction.html (2005) . Google Scholar