Evolution and Optimum Seeking

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52 Hill climbing Strategies (5) (0) (0) v 2 v 1 (0) (1) v 2 (2) (6) (1) (4) v 1 (1) (8) (7) (10) (2) v 2 (9) v (2) Figure 3.7: Strategy of Rosenbrock 1 (3) Starting point Success Failure Overall success (11)
Multidimensional Strategies 53 Numbering Iteration Test index/ Variable Step Remarks index iteration values lengths k nl + i x 1 x 2 s 1 s 2 (0) 0 0 0 9 2 2 starting point (1) 0 1 2 9 2 - success (2) 0 2 2 11 - 2 failure (3) 0 3 8 9 6 - failure (4) 0 4 2 8 - ;1 success (5) 0 5 ;1 8 ;3 - failure (6) 0 6 2 5 - ;3 failure (4) 1 0 2 8 2 2 transformation and orthogonalization (7) 1 1 3.8 7.1 2 - success (8) 1 2 2.9 5.3 - 2 success (9) 1 3 8.3 2.6 6 - success (10) 1 4 5.6 ;2.7 - 6 failure (11) 1 5 24.4 ;5.4 18 - failure (9) 2 0 8.3 2.6 2 2 transformation and orthogonalization In Figure 3.7, including the following table, a few iterations of the Rosenbrock strategy for n = 2 are represented geometrically. At the starting point x (00) the search directions are the same as the unit vectors. After three runs through (6 trials), the trial steps in each direction have led to a success followed by a failure. At the best condition thus attained, are generated. Five further trials (4) at x (04) = x (10) , new direction vectors v (1) 1 and v (1) 2 lead to the best point, (9) at x (13) = x (20) , of the second iteration, at which a new choice of directions is again made. The complete sequence of steps can be followed, if desired, with the help of the accompanying table. Numerical experiments show that within a few iterations the rotating coordinates become oriented such that one of the axes points along the gradient direction. The strategy thus allows sharp valleys in the topology of the objective function to be followed. Like the method of Hooke and Jeeves, Rosenbrock's procedure needs no information about partial derivatives and uses no line search method for exact location of relative minima. This makes it very robust. It has, however, one disadvantage compared to the direct pattern search: The orthogonalization procedure of Gram and Schmidt is very costly. It requires storage space of order O(n 2 ) for the matrices A = faijg and V = fvijg, and the number of computational operations even increases with O(n 3 ). At least in cases where the objective function call costs relatively little, the computation time for the orthogonalization with many variables becomes highly signi cant. Besides this, the number of parameters is in any case limited by the high storage space requirement. If there are constraints, care must be taken to ensure that the starting point is inside the allowed region and su ciently far from the boundaries. Examples of the application of
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Evolution and Optimum Seeking Hans-
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vi Multiple Data) machines with man
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viii 3.2.1.6 Complex Strategy of Bo
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Random Strategies 103 out, a limite
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Chapter 5 Evolution Strategies for
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The Two Membered Evolution Strategy
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A Multimembered Evolution Strategy
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Genetic Algorithms 151 too is the c
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Genetic Algorithms 153 mean objecti
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Genetic Algorithms 155 p( ∆x) 1 1
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Genetic Algorithms 157 out any chan
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Genetic Algorithms 159 Average Numb
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Simulated Annealing 161 There are t
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Tabu Search and Other Hybrid Concep
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Chapter 6 Comparison of Direct Sear
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Theoretical Results 167 Oettli, 196
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Theoretical Results 169 where 0
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Theoretical Results 171 tions. Simi
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Numerical Comparison of Strategies
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Core storage required 233 term \cor
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Chapter 7 Summary and Outlook So, i
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Summary and Outlook 237 numerical o
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Summary and Outlook 239 considerabl
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Summary and Outlook 241 is called t
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Summary and Outlook 243 to the prod
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Summary and Outlook 245 tition betw
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Summary and Outlook 247 the experim
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Chapter 8 References Glossary of ab
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References 251 Anscombe, F.J. (1959
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References 253 Bandler, J.W. (1969b
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References 255 Bendin, F. (1992), E
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References 257 Booth, A.D. (1955),
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References 259 Brent, R.P. (1971),
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References 261 Carroll, C.W. (1961)
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References 263 Courant, R., D. Hilb
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References 265 Davies, M., I.J. Whi
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References 267 Dvoretzky, A. (1956)
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References 269 Fiacco, A.V. (1974),
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References 271 Fogel, L.J., A.J. Ow
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References 273 Gill, P.E., W. Murra
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References 275 Graves, R.L., P. Wol
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References 277 Heckler, R., H.-P. S
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References 279 Ho meister, F., H.-P
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References 281 Idelsohn, J.M. (1964
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References 283 Karnopp, D.C. (1966)
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References 285 Kochen, M., H.M. Has
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References 287 Kunzi, H.P., H.G. Tz
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References 289 LeCam, L.M., J. Neym
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References 291 Mamen, R., D.Q. Mayn
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294 References Miele, A., H.Y. Huan
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296 References Murray, W. (Ed.) (19
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298 References Oldenburger, R. (Ed.
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300 References Peschel, M. (1980),
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302 References Powell, M.J.D. (1970
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304 References Rechenberg, I. (1989
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306 References Rybashov, M.V. (1969
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308 References Schmidt, J.W., K. Ve
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310 References Shedler, G.S. (1967)
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312 References Steinbuch, K. (1971)
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314 References Tabak, D. (1970), Ap
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316 References Varela, F.J., P. Bou
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318 References White, L.J., R.G. Da
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320 References Youden, W.J., O. Kem
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322 References Glossary of Abbrevia
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324 References
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326 Appendix A Problem 1.2 Objectiv
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328 Appendix A Minimum: x =(3 0:5)
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330 Appendix A Figure A.3: Graphica
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332 Appendix A Several of the searc
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338 Appendix A Minimum: Start: Prob
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340 Appendix A Problem 2.20 Objecti
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342 Appendix A Problem 2.23 Objecti
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344 Appendix A Start: x (0) i =10 f
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346 Appendix A on the interval 0 y
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348 Appendix A Problem 2.32 after B
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350 Appendix A Constraints: Gj(x) =
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352 Appendix A The constraints form
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354 Appendix A Start: x (0) =(250 2
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356 Appendix A Problem 2.44 As Prob
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358 Appendix A Figure A.29: Graphic
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360 Appendix A Start: Figure A.31:
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362 Appendix A Problem 3.2 (analogo
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364 Appendix A Problem 3.6 (analogo
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366 Appendix A Minimum: x i =0 for
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368 Appendix B LF (integer) Return
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370 Appendix B 4. Convergence crite
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372 Appendix B 8. Function Z(S,R) T
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374 Appendix B 25 L(K)=L(K+1) L(10)
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376 Appendix B LF=2 (continued) No
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378 Appendix B man), vol. 15 of Pro
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380 Appendix B instead of T, as a p
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382 Appendix B 15 CONTINUE 16 FF=F(
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384 Appendix B SK(KS)=AMAX1(SM(I),A
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386 Appendix B B.3 ( + ) Evolution
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388 Appendix B EPSILO (one dimensio
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390 Appendix B GLEICH (real functio
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392 Appendix B GLEICH, and TKONTR d
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394 Appendix B 1 NL=1+N-NS NM=N-1 N
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396 Appendix B ZSTERN=ZBEST K=LBEST
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398 Appendix B C NEGATIVE (LETHAL M
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400 Appendix B C C PREPARE FINAL DA
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402 Appendix B 2 IF(.NOT.BKOMMA.OR.
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404 Appendix B 32 WRITE(KANAL,126)
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406 Appendix B DO 4 I=1,NX KI=KI1 I
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408 Appendix B Subroutine MINMAX MI
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410 Appendix B 2 IF(U.LT..965487131
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412 Appendix B parameters by multip
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414 Appendix B
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416 Appendix C - SIMP Simplex strat
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418 Appendix C C.3.2 How to Install
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420 Appendix C { } return(0.26*(x[0
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422 Appendix C 1 No recombination 2
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424 Appendix C 8e-07 8e-07 Initial
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426 Index Banach, S., 10 Bandler, J
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428 Index Coordinate strategy, 41{4
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430 Index Evolution strategy, 3, 6,
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432 Index Graves, R.L., 23 Great de
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434 Index Khurgin, Ya.I., 89 Kiefer
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436 Index Michie, D., 102 Mickey, M
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438 Index Parkinson, J.M., 61 Parta
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440 Index Rotating coordinates meth
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442 Index Storey, C., 23, 50, 54 St
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444 Index Wilson, S.W., 103 Witt, U
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