Evolution and Optimum Seeking

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Test Problems for the Second Part of the Strategy Comparison 329 There are in nitely many local minima, the position of which can be speci ed by a transcendental equation: q q jx j = 2 tan ( jx j) For jx j 1wehave approximately and x ' ( (0:5+k)) 2 for k =1 2 3:::integer F (x ) 'jx j Whereas in reality none of the nite local minima is at the same time a global minimum, the nite word length of the digital computer used together with the system-speci c method of evaluating the sine function give rise to an apparent global minimum at x = 4:44487453 10 16 F (x )=;4:44487453 10 16 Counting from the origin it is the 67 108 864th local minimum in each direction. If x is increased above this value, the objective function value is always set to zero. (Note that this behavior is machine dependent.) Start: x (0) =0 F (x (0) )=0 Most strategies located the rst or highest local minimum left or right of the starting point (the origin). Depending on the sequence of random numbers, the two membered evolution method found (for example) the 2nd, 9th, and 34th local minimum. Only the (10, 100) evolution strategy almost always reached the apparent global minimum. Problem 2.4 Objective function: Minimum: Start: F (x) = nX i=1 Problem 2.5 after Booth (1949) Objective function: x 1 ; x 2 i 2 +(xi ; 1] 2 for n =5 x i =1 for i = 1(1)n F (x )=0 x (0) i =10 for i = 1(1)n F (x (0) ) = 40905 F (x) =(x 1 +2x 2 ; 7) 2 +(2x 1 + x 2 ; 5) 2
330 Appendix A Figure A.3: Graphical representation of Problem 2.4 for n =2 F (x) ==10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 = This minimum problem is equivalent to solving the following pair of linear equations: x 1 +2x 2 =7 2 x 1 + x 2 =5 Figure A.4: Graphical representation of Problem 2.5 F (x) ==1 9 25 49 81 121 169 225=
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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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Multidimensional Strategies 81 Brow
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Multidimensional Strategies 83 (197
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Multidimensional Strategies 85 meth
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Chapter 4 Random Strategies One gro
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Random Strategies 89 parts is taken
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Random Strategies 91 Pardalos and R
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Random Strategies 93 to write the n
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Random Strategies 95 the expectatio
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Random Strategies 97 maintained at
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Random Strategies 99 works entirely
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Random Strategies 101 the principle
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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
Page 285 and 286: References 277 Heckler, R., H.-P. S
Page 287 and 288: References 279 Ho meister, F., H.-P
Page 289 and 290: References 281 Idelsohn, J.M. (1964
Page 291 and 292: References 283 Karnopp, D.C. (1966)
Page 293 and 294: References 285 Kochen, M., H.M. Has
Page 295 and 296: References 287 Kunzi, H.P., H.G. Tz
Page 297 and 298: References 289 LeCam, L.M., J. Neym
Page 299 and 300: References 291 Mamen, R., D.Q. Mayn
Page 301 and 302: 294 References Miele, A., H.Y. Huan
Page 303 and 304: 296 References Murray, W. (Ed.) (19
Page 305 and 306: 298 References Oldenburger, R. (Ed.
Page 307 and 308: 300 References Peschel, M. (1980),
Page 309 and 310: 302 References Powell, M.J.D. (1970
Page 311 and 312: 304 References Rechenberg, I. (1989
Page 313 and 314: 306 References Rybashov, M.V. (1969
Page 315 and 316: 308 References Schmidt, J.W., K. Ve
Page 317 and 318: 310 References Shedler, G.S. (1967)
Page 319 and 320: 312 References Steinbuch, K. (1971)
Page 321 and 322: 314 References Tabak, D. (1970), Ap
Page 323 and 324: 316 References Varela, F.J., P. Bou
Page 325 and 326: 318 References White, L.J., R.G. Da
Page 327 and 328: 320 References Youden, W.J., O. Kem
Page 329 and 330: 322 References Glossary of Abbrevia
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Page 333 and 334: 326 Appendix A Problem 1.2 Objectiv
Page 335: 328 Appendix A Minimum: x =(3 0:5)
Page 339 and 340: 332 Appendix A Several of the searc
Page 341 and 342: 334 Appendix A Figure A.8: Graphica
Page 343 and 344: 336 Appendix A Figure A.9: Graphica
Page 345 and 346: 338 Appendix A Minimum: Start: Prob
Page 347 and 348: 340 Appendix A Problem 2.20 Objecti
Page 349 and 350: 342 Appendix A Problem 2.23 Objecti
Page 351 and 352: 344 Appendix A Start: x (0) i =10 f
Page 353 and 354: 346 Appendix A on the interval 0 y
Page 355 and 356: 348 Appendix A Problem 2.32 after B
Page 357 and 358: 350 Appendix A Constraints: Gj(x) =
Page 359 and 360: 352 Appendix A The constraints form
Page 361 and 362: 354 Appendix A Start: x (0) =(250 2
Page 363 and 364: 356 Appendix A Problem 2.44 As Prob
Page 365 and 366: 358 Appendix A Figure A.29: Graphic
Page 367 and 368: 360 Appendix A Start: Figure A.31:
Page 369 and 370: 362 Appendix A Problem 3.2 (analogo
Page 371 and 372: 364 Appendix A Problem 3.6 (analogo
Page 373 and 374: 366 Appendix A Minimum: x i =0 for
Page 375 and 376: 368 Appendix B LF (integer) Return
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Page 381 and 382: 374 Appendix B 25 L(K)=L(K+1) L(10)
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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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Evolution and Optimum Seeking

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