Evolution and Optimum Seeking

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178 Comparison of Direct Search Strategies for Parameter Optimization Besides their considerable cost in programming and computation time, numerical strategy comparisons entail further di culties. The e ectiveness of a method can be strongly in uenced by small programming details. A number of methods were not fully worked out by their originators and require heuristic rules to be introduced before they can be applied. The way in which this degree of freedom is exercised to de ne the procedure depends on the skill and experience of the programmer, which leads to large discrepancies between the results of investigations and the judgements of di erent authors on one and the same strategy. We have therefore, as far as possible, used already published programs (FORTRAN or ALGOL) for the algorithms or parts of them under study: One dimensional search with the Fibonacci method of Kiefer: M. C. Pike, J. Pixner (1965) Algorithm 2, Fibonacci search J. Boothroyd (1965) Certi cation of Algorithm 2 M. C. Pike, I. D. Hill, Note on Algorithm 2 F. D. James (1967) One dimensional search with the golden section method of Kiefer: K. J. Overholt (1967) Algorithm 16, Gold Direct search (pattern search) of Hooke and Jeeves: A. F. Kaupe, Jr. (1963) Algorithm 178, direct search M. Bell, M. C. Pike (1966) Remark on Algorithm 178 R. DeVogelaere (1968) Remark on Algorithm 178 F. K. Tomlin, L. B. Smith (1969) Remark on Algorithm 178 L. B. Smith (1969) Remark on Algorithm 178 Orthogonalization method for the strategies of Rosenbrock and of Davies, Swann, and Campey: J. R. Palmer (1969) An improved procedure for orthogonalizing the search vectors in Rosenbrock's and Swann's direct search optimization methods Derivative-free method of conjugate directions of M. J. D. Powell: M. J. Hopper (1971) Harwell subroutine library. A catalogue of subroutines, from which subroutine VA04A, updated May 20, 1970 (received as a card deck). Variable metric method of Davidon, Fletcher, and Powell as formulated by Stewart: S. A. Lill (1970) Algorithm 46. A modi ed Davidon method for nding the minimum of a function, using di erence approximation for the derivatives.
Numerical Comparison of Strategies 179 S. A. Lill (1971) Note on Algorithm 46 Z. Kovacs (1971) Note on Algorithm 46 Some of the parameters a ecting the accuracy were altered, either because the small values de ned by the author could not be realized on the available computer or because the closest possible approach to the objective could not have been achieved with them. Simplex method of Nelder and Mead: R. O'Neill (1971) Algorithm AS 47, function minimization using a simplex procedure A complete program for the Rosenbrock strategy: M. Machura, A. Mulawa (1973) Algorithm 450, Rosenbrock function minimization This was not applied because it could only treat the unconstrained case. The same applies to the code for the complex method of M. J. Box: J. A. Richardson, J. L. Kuester (1973) Algorithm 454, the complex method for constrained optimization The part of the strategy that, when the starting point is not feasible seeks a basis in the feasible region, is not considered here. Whenever the procedures named were published in ALGOL they have been translated into FORTRAN. All the other optimization strategies not mentioned here have also been programmed in FORTRAN, with close reference to the original publications. If one wanted to repeat the test series today, amuch larger number of codes could be made use of from the book of More and Wright (1993). 6.3.3 Results of the Tests 6.3.3.1 First Test: Convergence Rates for a Quadratic Objective Function In the rst part of the numerical strategy comparison the theoretical predictions of convergence rates and Q-properties will be tested, or, where these are not available, experimental data will be supplied instead. For this purpose two quadratic objective functions are used (Appendix A, Sect. A.1). In the rst (Problem 1.1) the matrix of coe cients is diagonal with unit diagonal elements, i.e., a scalar matrix. This simplest of all quadratic problems is characterized by concentric contour lines or surfaces that can be represented or imagined as circles in the two parameter case, spheres in the three parameter case, and surfaces of hyperspheres in the general case. The same pattern of contours but with arbitrary monotonic variation in the objective function occurs in the sphere model for which the average rates of progress of the evolution strategies could be determined theoretically (Rechenberg, 1973 and Chap. 5 of this book). The second objective function (Problem 1.2) has a matrix of coe cients with all nonzero elements. It represents a full quadratic problem (except for the missing linear term)
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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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Page 135 and 136: A Multimembered Evolution Strategy
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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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