Abstract
In this paper, we consider the problem of minimizing a function in severalvariables which could be multimodal and may possess discontinuities. A newalgorithm for the problem based on the genetic technique is developed. Thealgorithm is hybrid in nature in the sense that it utilizes the genetictechnique to generate search directions, which are used in an optimizationscheme and is thus different from any other methods in the literature.The algorithm has been tested on the Rosenbrock valley functions in 2 and 4dimensions, and multimodal functions in 2 and 4 dimensions, which are of ahigh degree of difficulty. The results are compared with the Adaptive RandomSearch, and Simulated Annealing algorithms. The performance of the algorithmis also compared to recent global algorithms in terms of the number offunctional evaluations needed to obtain a global minimum and results show thatthe proposed algorithm is better than these algorithms on a set of standardtest problems. It seems that the proposed algorithm is efficient and robust.
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References
Bazaraa, M.S., H. Sherali, and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, second edition, John Wiley and Sons, New York, 1993.
Reklaitis, G.V., A. Ravindran, and K.M. Ragsdell, Engineering Optimization: Methods and Applications.John Wiley, New York, 1983.
Corana, A., M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the Simulated Annealing algorithm”, ACM Trans. Math. Softw.13(3), 1987, 262–280.
Hajela, P., “Genetic search–An approach to the nonconvex optimization problem”, AIAA J. 28(7), 1990, 1205–1210.
Pham, D.T., and Y. Yang, “Optimization of multi-modal discrete functions using genetic algorithms”, Proc. Instn. Mech. Engrs.207, 1993, 53–59.
Luenberger, D.G., Linear and Nonlinear Programming, Addison Wesley, Second Edition, USA, 1984.
Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning.Reading Mass, Addison-Wesley, 1989.
Lawrence, D. (Ed.), Handbook of Genetic Algorithms,Van Nostrand Reinhold, New York, 1991.
Rosenbrock, H., “An automatic method for finding the greatest or least value of a function”, Comput. J.3, 1960, 175–184.
Masri, S.F., G.A. Bekey, and F.B. Safford, “A global optimization algorithm using adaptive random search”, Appl. Math. Comput.7, 1980, 353–375.
Rinnooy Kan, A.H.G., and G.T. Timmer, “Stochastic methods for global optimization”, American Journal of Mathematics and Management Sciences4, 1984, 7–40.
Price, W.L., “A controlled random search procedure for global optimization”, in: L.C.W. Dixon and G.P. Szegö (Eds.), Towards Global Optimization 2, North Holland, Amsterdam, 1978, pp. 71–84.
Törn, A.A., “A search-clustering approach to global optimization”, in L.C.W. Dixon and G.P. Szegö (Eds.), Towards Global Optimization 2, North Holland, Amsterdam, 1978, pp. 49–62.
De Biase, L., and F. Frontini, “A stochastic method for global optimization: its structure and numerical performance”, in L.C.W. Dixon and G.P. Szegö (Eds.), Towards Global Optimization 2, North-Holland, Amsterdam, 1978, pp. 85–102.
Rinnooy Kan, A.H.G., and G.T. Timmer, “Stochastic global optimization methods. Part II: multi level methods”, Mathematical Programming39, 1987, 57–78.
Dekkers, A., and E. Aarts, “Global optimization and simulated annealing”, Mathematical Programming50, 1991, 367–393.
Aluffi-Pentini, F., V. Parisi, and F. Zirilli, “Global optimization and stochastic differential equations”, Journal of Optimization Theory and Applications47, 1985, 1–16.
Dixon, L.C.W., and G.P. Szegö, “The global optimization problem: an introduction”, in: L.C.W. Dixon and G.P. Szegö (Eds.), Towards Global Optimization 2, North Holland, Amsterdam, 1978, pp. 1–15.
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Hussain, M.F., Al-Sultan, K.S. A Hybrid Genetic Algorithm for Nonconvex Function Minimization. Journal of Global Optimization 11, 313–324 (1997). https://doi.org/10.1023/A:1008290611151
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DOI: https://doi.org/10.1023/A:1008290611151