Best Minimizing Algorithm for Shape-measure Method

Volume 5, Issue 3, pp 176 - 184 http://dx.doi.org/10.22436/jmcs.05.03.05

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Authors

Alireza Fakharzadeh - Department of Mathematics, Faculty of basic Sciences, Shiraz University of Technology, Shiraz, Iran. Zahra Rafiei - Department of Mathematics, Faculty of basic Sciences, Shiraz University of Technology, Shiraz, Iran.

Abstract

The Shape-Measure method for solving optimal shape design problems (OSD) in cartesian coordinates is divided into two steps. First, for a fixed shape (domain), the problem is transferred to the space of positive Radon measures and relaxed to a linear programming in which its optimal coefficients determine the optimal pair of trajectory and control. Then, a standard minimizing algorithm is used to identify the best shape. Here we deal with the best standard algorithm to identify the optimal solution for an OSD sample problem governed by an elliptic boundary control problem.

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Keywords

• elliptic equation
• Radon measure
• optimal shape
• search techniques.

MSC

•  49M37
•  68W25

References

• [1] O. Bozorg Haddad, A. Afshar, M. A. Marino, Honey-Bees Mating Optimization (HBMO) Algorithm:A New Heuristic Approach for Water Resource Optimization, Water Resources Management, 20 (2006), 661-680.


• [2] E. D. Dolan, Pattern Search Behaviour in Nonlinear Optimization, Phd Thesis from the College of William-Mary in Virginia , (1999)


• [3] A. E. Eiben, E. H. Aarts, K. M. VanHee , Global convergence of jenetic algorithms: A markov chain analysis, Department of Mathematics and Computing Science, Eindhoven University of Technology, springer, (1991), 4-12.


• [4] V. Esat, M. J. Hall, Water resources system optimization using genetic algorithms, in Hydroinformatics, Proc., 1st Int. Conf. on Hydroinformatics, Balkema, Rotterdam, Netherlands, (1994), 225-231.


• [5] A. Fakharzade, J. E. Rubio, Shape-measure method for solving elliptic optimal shape problem, Bulletin of the Iranian Math. Soc, 1 (2001), 41-63.


• [6] A. Fakharzade, J. E. Rubio, Best domain for an elliptic problem in cartesian coordinates by Means of shape-measure, AJOP Asian, J. of control, 11 (2009), 536-554.


• [7] D. B. Fogel, Evolving Artificial Intelligence, phddissert, San Diego: University of California (1992)


• [8] M. Gen, R. Cheng, Genetic Algorithm and Engineering Design, John Wiley and Sons, (1997)


• [9] V. Granville, M. Krivanek, J. P. Rasson, Simulated annealing-a proof of convergence, IEEE Transactions on Pattern Analysis and Machine Intelligence, 16 (1994), 652-656.


• [10] L. Han, M. Neumann, Effect of dimensionality on the Nelder-Mead simplex method, Optimization Methods and Software, 21 (2006), 1-16.


• [11] R. Hooke, T. A. Jeeves, Direct search solution of numerical and statistical problems, Journal of the Association for Computing Machinery, 8 (1961), 212-229.


• [12] T. G. Kolda, R. M. Lewis, V. J. Torczon, Optimization by direct search: New perspectives on some classical and modern methods, SIAM Review, 45 (2003), 385-482.


• [13] V. P. Mikhailov, Partial Differential Equations, MIR publisher, Moscow (1978)


• [14] J. A. Nelder, R. A. Mead, simplex method for function minimization, The Computer Journal, 7 (1965), 308-313.


• [15] J. Nocedal, S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, second ed., Springer, New York (2006)


• [16] J. E. Rubio, Control and Optimization: the linear treatment of nonlinear problems, Manchester University Press, Manchester (1986)


• [17] W. Rudin, Real and Complex Analysis, 2 Edition, Tata McGraw-Hill Publishing Co Ltd, new Dehli (1983)


• [18] M. Parlar, Interactive Operations Research with Maple, Birkhauser Boston, Springer Verlag, New York (2010)


• [19] V. J. Torczon, On the convergence of the multidirectional search algorithm, SIAM Journal on Optimization , 1 (1991), 123-145.


• [20] G. R. Walsh, Method of optimization, John wiley and Sons Ltd, (1975)


• [21] M. H. Wright , Direct search methods:Once scorned, now respectable, in: D.F. Griffiths, G.A. Watson (Eds.), Numerical Analysis 1995, Addison Wesley Longman, Harlow, U. K., (1996), 191-208.