The resolution of random numbers are a serious problem that Stochastic Sketching has yet been encountered. A uniform random number generator that is capable of gener-ating random numbers with better resolution is necessary when Stochastic Sketching is applied to relatively high-dimensional objective functions. Even though using space-filling curves to generalize Stochastic Sketching to solve multidimensional objec-tive functions is theoretically feasible, it may introduce some severe practical difficulties, such as the resolution of random numbers. Therefore, other approaches for generalization to high dimension form the one-dimensional method should be further studied. The line model is currently used as the sketching model in the implementation. Other models ful-filling the requirements of sketching models may be used. The parameter settings always pose a problem for random search methods and, hence, stochastic algorithms. Good initial parameters were reported in [31] based on previous computing experience with the algorithms. With this fixed parameters setting, the Stochastic Sketching method performs reasonably well. In future studies, we may try to let some of the parameters of Stochas-tic Sketching be adjusted automaStochas-tically by introducing the self-adaptation technique to Stochastic Sketching, which currently prevails in the field of Evolutionary Algorithms.
In summary, a new method based on the simulation of human behavior has been proposed for global optimization. All essential components of Stochastic Sketching have been introduced and discussed in detail as well as the background and concepts according to which Stochastic Sketching was designed and developed. The mathematical foundation of Stochastic Sketching is the Pincus theorem. Some multi-modal functions with good results. It seems that this method is comparable in solution quality and the number of function evaluations with the Evolution
Strate-( ,c cα,ζ ζ0, β,Ps)=(1500 3 25 5 0 50 0 2, . , , . , . ),
gies method and is better than various variants of the genetic algorithms. The calculation in-volved in each step for Stochastic Sketching is less than that for Evolution Strategies.
Table 3. Comparison with evolutionary programming.
Stochastic Sketching Evolutional Programming Function
Succ. Rate Avg. Eval. Succ. Rate Avg. Eval.
f1 1.0 126.67 1.0 305.00
f2 1.0 4050.41 0.9 1952.50
REFERENCES
1. A. Torn and A. Zilinskas, Global Optimization, Vol. 350 of LNCS, Springer-Verlag, New York, USA; Berlin, 1989.
2. A. V. Levy and A. Montalvo, “The tunneling algorithm for the global minimization of functions,” SIAM Journal on Scientific Computing, Vol. 6, No. 1, 1985, pp.
15-29.
3. T. Back, Evolutionary Algorithms in Theory and Practice, Oxford University Press, New York, USA, 1996.
4. L. J. Fogel, A. J. Owens, and M. J. Walsh, Artificial Intelligence through Simulated Evolution, Wiley, New York, 1966.
5. K. A. D. Jong, An Analysis of the Behavior of a Class of Genetic Adaptive Systems, Ph.D. thesis, University of Michigan, 1975.
6. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley Publishing Company, Inc., Reading, MA, USA, 1989.
7. J. D. Babley, “The behavior of adaptive systems which employ genetic and correla-tion algorithms,” Ph.D. thesis, University of Michigan, 1967.
8. R. S. Rosenberg, “Simulation of genetic populations with biochemical properties,”
Ph.D. thesis, University of Michigan, 1967.
9. R. Weinberg, “Computer simulation of a living cell,” Ph.D. thesis, University of Michigan, 1970.
10. D. J. Cavicchio, “Adaptive search using simulated evolution,” Unpublished doctoral dissertation, University of Michigan, 1970.
11. R. B. Hollstien, “Artificial genetic adaptation in computer control systems,” Ph.D.
thesis, University of Michigan, 1971.
12. J. D. Schaffer, “Multiple objective optimization with vector evaluated genetic algo-rithms,” in Proceedings of the First International Conference on Genetic Algorithms and Their Applications, pp. 93-100.
13. D. E. Goldberg, “Genetic algorithms with sharing for multimodal function optimiza-tion,” in Grefenstette [14], pp. 41-49.
14. D. E. Goldberg and R. E. Smith, “Nonstationary function optimization using genetic algorithms with dominance and diploidy,” Genetic Algorithms and Their Applica-tions: Proceeding of the Second International Conference on Genetic Algorithms, 1987, pp. 59-68.
15. K. Deb and D. E. Goldberg, “An investigation of niche and species formation in ge-netic function optimization,” in Schaffer [16], pp. 42-50.
16. J. D. Schaffer, R. A. Caruana, L. J Eshelman, and R. Das, “A study of control
pa-rameters affecting online performance of genetic algorithms for function optimiza-tion,” in Proceedings of the Third International Conference on Genetic Algorithms, pp. 51-60.
17. M. F. Bramlette, “Initialization, mutation and selection methods in genetic algo-rithms for function optimization,” in Belew and Booker [19], pp. 100-107.
18. M. Muselli and S. Ridella, “Global optimization of functions with the interval ge-netic algorithm,” Complex System, Vol. 6, No. 3, 1992, pp. 193-212.
19. W. E. Hart and R. K. Belew, “Optimizing an arbitrary function is hard for the genetic algorithm,” in Proceedings of the Fourth International Conference on Genetic Algo-rithms, pp. 190-195.
20. V. S. Gordon and D. Whitley, “Serial and parallel genetic algorithms as function optimizers,” in Forrest [39], pp. 177-183.
21. D. Powell and M. M. Skolnick, “Using genetic algorithms in engineering design op-timization with non-linear constraints,” in Forrest [39], pp. 424-431
22. J. Arabas, J. M. Mulawka, and J. Pokrasniewicz, “A new class of the crossover op-erators for the numerical optimization,” in Eshelman [23], pp. 42-48.
23. Z. Michalewica, “Genetic algorithms, numerical optimization, and constraints,” in Proceedings of the Sixth International Conference on Genetic Algorithms, pp.
151-158.
24. S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, “Optimization by simulated an-nealing,” Science, Vol. 220, No. 4598, 1983, pp. 671-680.
25. V. Cerny, “Thermodynamical approach to the traveling salesman problem: an effi-cient simulation algorithm,” Journal of Optimization Theory and Application, Vol.
45, 1985, pp. 41-51.
26. G. H. Sasaki and B. Hajek, “The time complexity of maximum matching by simu-lated annealing,” Journal of ACM, Vol. 35, No. 2, 1988, pp. 387-403.
27. U. Faigle and R. Schrader, “On the convergence of stationary distributions in simu-lated annealing algorithms,” Information Processing Letters, Vol. 27, No. 4, 1988, pp. 189-194.
28. F. Aluffi-Pentini, V. Parisi, and F. Zirille, “Global optimization and stochastic dif-ferential equations,” Journal of Optimization Theory and Application, Vol. 47, No. 1, 1985, pp. 1-16.
29. A. Coranna, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal func-tions of continuous variables with the simulated annealing algorithm,” ACM Trans-actions on Mathematical Software, Vol. 13, No. 3, 1987, pp. 262-280.
30. Y. P. Chen, “Stochastic sketching: A new method for global optimization,” Master’s thesis, Dept. of CSIE, National Taiwan University, 1997.
31. Y. P. Chen, J. T. Horng, and C. Y. Kao, “Stochastic sketching: A new method for global optimization,” Soft Computing, Vol. 3, Issue 2, 1999, pp. 101-110.
32. H. Sagan, Space-Filling Curves, Springer-Verlag, New York, USA, Berlin, 1994.
33. St. C. Mine, “Peano curves and smoothness of functions,” Advances in Mathematics, Vol. 35, 1980, pp. 129-157.
34. M. Pincus, “A closed form solution of certain programming problems,” Operations Research, Vol. 16, No. 3, 1968, pp. 690-694.
35. G. B. Thomas, Jr. and R. L. Finney, Calculus and Analytic Geometry, Addi-son-Wesley Publishing Company, Inc., Reading, MA, USA, 1996.
36. A. A. Zhigljavsky, Theory of Global Random Search, Mathematics and Its Applica-tions, Vol. 65, Kluwer Academic Publishers, Norwell, Massachusetts, USA, 1991.
37. B. C. Cetin, J. Barhen, and J. W. Burdick, “Terminal repeller unconstrained suben-ergy tunneling (TRUST) for fast global optimization,” Journal of Optimization The-ory and Applications, Vol. 77, No. 1, 1993, pp. 97-126.
38. G. H. Koon and A. V. Sebald, “Some interesting test functions for evaluation evolu-tionary programming strategies,” in Proceedings of the Fourth Annual Conference on Evolutionary Programming, 1995, pp. 479-499.
39. L. J. Eshelman and J. D. Schaffer, Crossover’s Niche, in Proceedings of the Fifth International Conference on Genetic Algorithms, pp. 9-14. University of Illinois at Urbana-Champaign, July 17-21, 1993. Office of Naval Research, Naval Research Laboratory, Philips Laboratory, North American Philips Corporation and Interna-tional Society for Genetic Algorithms, Morgan Kaufmann Publishers, Inc.
6. APPENDIX
Table 4. Notations of stochastic sketching.
Notation Description
s(· ) sketching function
Pf,c(· ) sampling guide for a given f(· ) and c N0 number of initial sampling points
c zooming controller
cα decreasing factor of the zooming controller ζ precision threshold
ζβ decreasing rate of the precision threshold Ps satisfaction probability
Table 5. Other notation.
Notation Description
µ(· ) Lebesgue measure function Prob(· ) probability function
φ empty set
B {0,1}
E R with the Euclidean norm
A feasible region of the objective function x, y,… scalars
Xˆ Yˆ,… n-dimensional vectors 0ˆ zero vector (0,0,…,0)
f* global optimum
x*, Xˆ* global optimum point
f local optimum
x*, Xˆ* local optimum point
f* optimum reported by an algorithm x+, Xˆ+ optimum point reported by an algorithm
Jorng-Tzong Horng (¹Òà) was born in Nantou, Taiwan, on April 10, 1960. He received the Ph.D. degree in Computer Science and Information Engineering from Na-tional Taiwan University, Taipei, Taiwan, in April 1993. He is currently an Associate Professor of the Department of Computer Science and Information Engineering at Na-tional Central University, Chung-Li, Taiwan. His current research interests include ob-ject-oriented database systems, distributed database systems, query processing and opti-mization, genetic algorithms, and bioinformatics.
Ying-Ping Chen (W) received both the B.S. degree and the M.S degree in Computer Science and Information Engineering from the National Taiwan University in 1995 and 1997, respectively. From 1999, he is a PhD student in the Department of Com-puter Science at the University of Illinois at Urbana-Champaign. His research interests are in the field of Genetic Algorithms and Evolutionary Computation.
Cheng-Yan Kao (=) was born in Taipei, Taiwan, 1948. He received B.S. in mathematics from National Taiwan University, Taipei, Taiwan, in 1971, and the M.S.
degree in computer science in 1976, the M.S. degree in statistics in 1978, and the Ph.D.
degree in computer science in 1981, all from the University of Wisconsin-Madison. He worked for Ford Aerospace, the Unisys Corporation, and worked for General Electric from 1980 to 1989 at the Johnson Space Center, NASA, Houston, TX. He has been a professor with the Department of Computer Science and Information Engineering, Na-tional Taiwan University since 1990. He has published more than 40 technical papers in various journals and conference records. His research interests include evolutionary computation, bioinformatics, optimization, and artificial intelligence.