Evolution and Optimum Seeking

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vi Multiple Data) machines with many processors working in parallel for one task seem to wait for inherently parallel problem solving concepts like ESs. Biological metaphors prevail within the new branch of Arti cial Intelligence, called Arti cial Life (AL). Third, updating this dissertation from 1974/1975 once more (after adding only a few pages to Chapter 7 in 1981) can be done without rewriting the bulk of the chapters on traditional approaches. Since the emphasis always has been centered on derivativefree direct optimum-seeking methods, it should be su cient to add material on three concepts now, i.e., Genetic Algorithms (GAs), Simulated Annealing (SA), and Tabu Search (TS). This was done with the new Sections 5.3 to 5.5 in Chapter 5. Another innovation is a oppy disk with all those procedures which had been used for the test series in the 1970s, along with a users' manual. Hopefully, some incorrectnesses have been deleted now, too. A rst thank goes again to my friend Dr. Mike Finnis whose translation of my German original into English still forms the core of this book. Thanks go also to those who helped me in completing this update, especially Ms. Heike Bracklo, who brought the scanned ASCII text into LaTeX formats, Mr. Ulrich Hermes, Mr. Jorn Mehnen, and Mr. Joachim Sprave forthemany graphs and ready for use computer programs, as well as all those who helped in the process of proofreading the complete work. Finally, Iwould like to thank the Wiley team for the fruitful collaboration during the process of editing the camera-ready script. Dortmund, Autumn 1994 Hans-Paul Schwefel
Contents Preface v 1 Introduction 1 2 Problems and Methods of Optimization 5 2.1 General Statement of the Problems : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Particular Problems and Methods of Solution : : : : : : : : : : : : : : : : 6 2.2.1 Experimental Versus Mathematical Optimization : : : : : : : : : : 6 2.2.2 Static Versus Dynamic Optimization : : : : : : : : : : : : : : : : : 9 2.2.3 Parameter Versus Functional Optimization : : : : : : : : : : : : : : 10 2.2.4 Direct (Numerical) Versus Indirect (Analytic) Optimization : : : : : : : : : : : : : : : : : : : 13 2.2.5 Constrained Versus Unconstrained Optimization : : : : : : : : : : : 16 2.3 Other Special Cases : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 3 Hill climbing Strategies 23 3.1 One Dimensional Strategies : : : : : : : : : : : : : : : : : : : : : : : : : : 25 3.1.1 Simultaneous Methods : : : : : : : : : : : : : : : : : : : : : : : : : 26 3.1.2 Sequential Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.1.2.1 Boxing in the Minimum : : : : : : : : : : : : : : : : : : : 28 3.1.2.2 Interval Division Methods : : : : : : : : : : : : : : : : : : 29 3.1.2.2.1 Fibonacci Division. : : : : : : : : : : : : : : : : : 29 3.1.2.2.2 The Golden Section. : : : : : : : : : : : : : : : : 32 3.1.2.3 Interpolation Methods : : : : : : : : : : : : : : : : : : : : 33 3.1.2.3.1 Regula Falsi Iteration. : : : : : : : : : : : : : : : 34 3.1.2.3.2 Newton-Raphson Iteration. : : : : : : : : : : : : 35 3.1.2.3.3 Lagrangian Interpolation. : : : : : : : : : : : : : 35 3.1.2.3.4 Hermitian Interpolation. : : : : : : : : : : : : : : 37 3.2 Multidimensional Strategies : : : : : : : : : : : : : : : : : : : : : : : : : : 38 3.2.1 Direct Search Strategies : : : : : : : : : : : : : : : : : : : : : : : : 40 3.2.1.1 Coordinate Strategy : : : : : : : : : : : : : : : : : : : : : 41 3.2.1.2 Strategy of Hooke andJeeves: Pattern Search : : : : : : : 44 3.2.1.3 Strategy of Rosenbrock: Rotating Coordinates : : : : : : : 48 3.2.1.4 Strategy of Davies, Swann, and Campey (DSC) : : : : : : 54 3.2.1.5 Simplex Strategy of Nelder and Mead : : : : : : : : : : : 57 vii
Page 1: Evolution and Optimum Seeking Hans-
Page 6 and 7: viii 3.2.1.6 Complex Strategy of Bo
Page 9 and 10: Chapter 1 Introduction There is sca
Page 11 and 12: Introduction 3 simple matter to col
Page 13 and 14: Chapter 2 Problems and Methods of O
Page 15 and 16: Particular Problems and Methods of
Page 27 and 28: Other Special Cases 19 with expecta
Page 29 and 30: Other Special Cases 21 parts of the
Page 31 and 32: Chapter 3 Hill climbing Strategies
Page 33 and 34: One Dimensional Strategies 25 then
Page 35 and 36: One Dimensional Strategies 27 Even
Page 37 and 38: One Dimensional Strategies 29 The b
Page 39 and 40: One Dimensional Strategies 31 F(x)
Page 41 and 42: One Dimensional Strategies 33 For t
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Page 45 and 46: One Dimensional Strategies 37 F P F
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Multidimensional Strategies 47 Numb
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Multidimensional Strategies 49 wher
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Multidimensional Strategies 51 Step
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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
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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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