Evolution and Optimum Seeking

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72 Hill climbing Strategies Step 1: (Specify rst set of directions) Set v (0) i = ei for all i = 1(1)n and set k =0. Step 2: (Start outer loop) Set x (k0) = x (k) and i =1. Step 3: (Line search) Determine x (ki) such that F (x (ki) )=minfF (x s (ki;1) + sv (k) i )g: Step 4: (Inner loop) If i
Multidimensional Strategies 73 otherwise set x (0) = y (3) v (0) 1 v (0) i = v (k) i = y (3) ; y (1) for i = 2(1)n, k = 0, and go to step 2. Figure 3.9 illustrates a few iterations for a hypothetical two parameter function. Each of the rst loops consists of n +1 = 3 line searches and leads to the adoption of a new search direction. If the objective function had been of second order, the minimum would certainly have been found by the last line search of the second loop. In the third and fourth loops it has been assumed that the trial steps have led to a decision not to exchange directions, thus the old direction vectors, numbered v 3 and v 4 are retained. Further loops, e.g., according to step 9, are omitted. The quality of the line searches has a strong in uence on the construction of the conjugate directions. Powell uses a sequence of Lagrangian quadratic interpolations. It is terminated as soon as the required accuracy is reached. For the rst minimization within an iteration three points and Equation (3.16) are used. The argument values taken in direction vi are: x (the starting point), x + si vi, and either x +2si vi or x ; si vi, according to whether F (x + si vi)
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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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Chapter 1 Introduction There is sca
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Introduction 3 simple matter to col
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Chapter 2 Problems and Methods of O
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Particular Problems and Methods of
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Other Special Cases 19 with expecta
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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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