Evolution and Optimum Seeking

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130 Evolution Strategies for Numerical Optimization This nally gives the probability function for the useful distance s0 covered in one generation, expressed in units of u: w(s 0 )= a rE e ; a au2 ; 2 u e 0 2 I ;1(au) @1 ; ae ; a 2 Zu v=0 av2 ; v e 2 I ;1(av) dv Since the expectation value of this distribution is not readily obtainable, we shall determine its maximum to give an approximation ~'. From the necessary condition with the more concise notation we obtain the relation =1+ @D(u) @u u=1; ~'=rE @w(s 0 ) @s 0 s 0 =~' ! =0 D(y) =ae ; a ay2 ; 2 y e 2 I ;1(ay) 0 B [D(1 ; ~'=rE)] ;2 @1 ; Z 1; ~'=rE v=0 1 C 1 A ;1 D(v) dvA (5.18) Except for the upper limit of integration, this is the same integral that made it so di cult to obtain the exact solution for the rate of progress in the (1+1) evolution strategy (see Rechenberg, 1973). Under the condition 1and =a 1, which means for many variables and at a large enough distance from the optimum, Rechenberg obtained an estimate by expanding Debye's asymptotic series representation of the Bessel function (e.g., Jahnke-Emde-Losch, 1966) in powers of =a. Without giving here the individual steps in the derivation, the result is Z1 D(v) dv ' 1 " 1 ; erf 2 !# p + 2 a 2 p a p 2 v=0 " exp ; ( ; 1)2 ! ; exp 2 a 2 ; 2 a !# (5.19) It is clear from Equation (5.4) that the rate of progress of the (1+1) strategy for the two membered evolution varies inversely as the number of variables. Even if a higher convergence rate is expected from the multimembered scheme, with many descendants per parent, there will be no change in the relation to n, thenumber of parameters. In addition to the assumptions already made regarding and =a, without further risk to the validity of the approximate theory we can assume that 1 ; ~'=rE ' 1. Equation (5.19) can now also be applied here. For the partial di erential @D(u) @u u=1; ~'=rE we obtain with the use of the Debye series again: @D(u) @u u=1; ~'=rE = D(1 ; ~'=rE) " a exp a (1 ; ~'=rE) ! 1 + 1 ; ~'=rE # ; a (1 ; ~'=rE)
A Multimembered Evolution Strategy 131 Figure 5.9: Rate of progress for the sphere model
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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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Other Special Cases 21 parts of the
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Chapter 3 Hill climbing Strategies
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One Dimensional Strategies 25 then
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One Dimensional Strategies 27 Even
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One Dimensional Strategies 29 The b
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One Dimensional Strategies 31 F(x)
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One Dimensional Strategies 33 For t
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One Dimensional Strategies 35 It is
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One Dimensional Strategies 37 F P F
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Multidimensional Strategies 39 the
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Multidimensional Strategies 41 asch
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Multidimensional Strategies 43 e 2
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Multidimensional Strategies 45 Step
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Multidimensional Strategies 47 Numb
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Multidimensional Strategies 49 wher
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Multidimensional Strategies 53 Numb
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Multidimensional Strategies 55 The
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Multidimensional Strategies 57 Anum
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Multidimensional Strategies 61 Iter
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Multidimensional Strategies 63 (If
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Multidimensional Strategies 65 eval
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Multidimensional Strategies 67 The
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Multidimensional Strategies 69 devi
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Multidimensional Strategies 71 spec
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Multidimensional Strategies 73 othe
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Multidimensional Strategies 75 anot
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Multidimensional Strategies 77 3.2.
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Page 107 and 108: Random Strategies 99 works entirely
Page 109 and 110: Random Strategies 101 the principle
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Page 113 and 114: Chapter 5 Evolution Strategies for
Page 115 and 116: The Two Membered Evolution Strategy
Page 127 and 128: A Multimembered Evolution Strategy
Page 137: A Multimembered Evolution Strategy
Page 159 and 160: Genetic Algorithms 151 too is the c
Page 161 and 162: Genetic Algorithms 153 mean objecti
Page 163 and 164: Genetic Algorithms 155 p( ∆x) 1 1
Page 165 and 166: Genetic Algorithms 157 out any chan
Page 167 and 168: Genetic Algorithms 159 Average Numb
Page 169 and 170: Simulated Annealing 161 There are t
Page 171 and 172: Tabu Search and Other Hybrid Concep
Page 173 and 174: Chapter 6 Comparison of Direct Sear
Page 175 and 176: Theoretical Results 167 Oettli, 196
Page 177 and 178: Theoretical Results 169 where 0
Page 179 and 180: Theoretical Results 171 tions. Simi
Page 181 and 182: Numerical Comparison of Strategies
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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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