Evolution and Optimum Seeking

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20 Problems and Methods of Optimization This means that the work step length goes to zero very much faster than the test step length, in order to compensate for the growing in uence of the perturbations. Blum (1954b) and Dvoretzky (1956) describe how to apply this process to multidimensional problems. The increment in the objective function, hence an approximation to the gradient vector, is obtained from n +1 observations. Sacks (1958) uses 2 n trial steps. The stochastic approximation can thus be regarded, in a sense, as a particular gradient method. Yet other basic strategies have been proposed these adopt only the choice of step lengths from the stochastic approximation, while the directions are governed by other criteria. Thomas and Wilde (1964) for example, combine the stochastic approximation with the relaxation method of Southwell (1940, 1946). Kushner (1963) and Schmitt (1969) even take random directions into consideration. All the proofs of convergence of the stochastic approximation assume unimodal objective functions. A further disadvantage is that stability against perturbations is bought atavery high cost, especially if the number of variables is large. How many steps are required to achieve a given accuracy can only be stated if the probability density distribution of the stochastic perturbations is known. Many authors have tried to devise methods in which the basic procedure can be accelerated: e.g., Kesten (1958), who only reduces the step lengths after a change in direction of the search, or Odell (1961), who makes the lengths of the work steps dependent on measured values of the objective function. Other attempts are directed towards reducing the e ect of the perturbations (Venter, 1967 Fabian, 1967), for example by making only the direction and not the size of the gradients determine the step lengths. Bertram (1960) describes various examples of applications. More of such work is that of Krasulina (1972) and Engelhardt (1973). In this introduction many classes of possible or practically occurring optimization problems and methods have been sketched brie y, but the coverage is far from complete. No mention has been made, for example, of broken rational programming, norofgraphical methods of solution. In operations research especially (Henn and Kunzi, 1968) there are many special techniques for solving transport, allocation, routing, queuing, andwarehouse problems, such as network planning and other graph theoretical methods. This excursion into the vast realm of optimization problems was undertaken because some of the algorithms to be studied in more depth in what follows, especially the random methods of Chapter 4, owe their origin and nomenclature to other elds. It should also be seen to what extent methods of direct parameter optimization permeate the other branches of the subject, and how they are related to each other. An overall scheme of how thevarious branches are interrelated can be found in Saaty (1970). If there are two ormoreobjectives at the same time and occasion, and especially if these are not con ict-free, single solution points in the decision variable space can no longer give the full answer to an optimization question, not even in the otherwise simplest situation. How to look for the whole subset of e cient, non-dominated, or Paretooptimal solutions can be found under keywords like vector optimization, polyoptimization or multiple criteria decision making (MCDM) (e.g., Bell, Keeney, and Rai a, 1977 Hwang and Masud, 1979 Peschel, 1980 Grauer, Lewandowski, and Wierzbicki, 1982 Steuer, 1986). Game theory comes into play when several decision makers have access to di erent
Other Special Cases 21 parts of the decision variable set only (e.g., Luce and Rai a, 1957 Maynard Smith, 1982 Axelrod, 1984 Sigmund, 1993). No consideration is given here to these special elds.
Page 1: Evolution and Optimum Seeking Hans-
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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 79 Step
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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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