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Global+Optimization+Algorithms+Theory+and+Application_Part15

# Global+Optimization+Algorithms+Theory+and+Application_Part15...

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15.5 General Information 281 Function Optimization [2042] 15.5.2 Online Resources Some general, online available ressources on Memetic Algorithms are: http://www.densis.fee.unicamp.br/ ~ moscato/memetic_home.html [accessed 2008-04-03] Last update: 2002-08-16 Description: The Memetic Algorithms’ Home Page by Pablo Moscato 15.5.3 Books Some books about (or including significant information about) Memetic Algorithms are: Hart, Krasnogor, and Smith [901]: Recent Advances in Memetic Algorithms Corne, Dorigo, Glover, Dasgupta, Moscato, Poli, and Price [448]: New Ideas in Optimisation Glover and Kochenberger [813]: Handbook of Metaheuristics Grosan, Abraham, and Ishibuchi [862]: Hybrid Evolutionary Algorithms

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16 Downhill Simplex (Nelder and Mead) 16.1 Introduction The downhill simplex 1 (or Nelder-Mead method or amoeba algorithm 2 ) published by Nelder and Mead [1517] in 1965 is an single-objective optimization approach for searching the space of n -dimensional real vectors ( G R n ) [1561, 1230]. Historically, it is closely related to the simplex extension by Spendley et al. [1941] to the Evolutionary Operation method mentioned in Section 2.1.6 on page 101 [1276]. Since it only uses the values of the objective functions without any derivative information (explicit or implicit), it falls into the general class of direct search methods [2260, 2054], as most of the optimization approaches discussed in this book do. Downhill simplex optimization uses n +1 points in the R n . These points form a polytope 3 , a generalization of a polygone, in the n -dimensional space – a line segment in R 1 , a triangle in R 2 , a tetrahedron in R 3 , and so on. Nondegenerated simplexes, i.e., those where the set of edges adjacent to any vertex form a basis in the R n , have one important festure: The result of replacing a vertex with its reﬂection through the opposite face is again, a nondegenerated simplex (see Fig. 16.1.a ). The goal of downhill simplex optimization is to replace the best vertex of the simplex with an even better one or to ascertain that it is a candidate for the global optimum [1276]. Therefore, its other points are constantly ﬂipped around in an intelligent manner as we will outline in Section 16.3 . Like hill climbing approaches, the downhill simplex may not converge to the global minimum and can get stuck at local optima [1230, 1383, 2046]. Random restarts (as in Hill Climbing with Random Restarts discussed in Section 10.5 on page 256 ) can be helpful here. 16.2 General Information 16.2.1 Areas Of Application Some example areas of application of downhill simplex are: 1 http://en.wikipedia.org/wiki/Nelder-Mead_method [accessed 2008-06-14] 2 In the book Numerical Recipes in C++ by Press et al. [1675], this optimization method is called “amoeba algorithm”. 3 http://en.wikipedia.org/wiki/Polytope [accessed 2008-06-14]

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284 16 Downhill Simplex (Nelder and Mead) Application References Chemistry, Chemical Engineering [2142, 1401, 2127, 145] Robotics [371] Physics [1604, 1493] Biochemistry [2293] Data Mining and Data Analysis [1812] 16.2.2 Online Resources Some general, online available ressources on downhill simplex are: http://math.fullerton.edu/mathews/n2003/NelderMeadMod.html [accessed 2008-06-14] Last update: 2004-07-22 Description:
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