Evolution and Optimum Seeking

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66 Hill climbing Strategies A distinction is sometimes drawn between short step methods, which evaluate the gradients again after a small step in the direction rF (x (k) ) (for maximization) or ;rF (x (k) ) (for minimization), and their equivalent long step methods. Since for nite step lengths s (k) it is not certain whether the new variable vector is really better than the old, after the step the value of the objective function must be tested again. Working with small steps increases the number of objective function calls and gradientevaluations. Besides F (x) n partial derivatives must be evaluated. Even if the slopes can be obtained analytically and can be speci ed as functions, there is no reason to suppose that the number of computational operations per function call is much less than for the objective function itself. Except in special cases, the total cost increases roughly as the weighting factor (n +1) and the number of objective function calls. This also holds if the partial derivatives are approximated by di erential quotients obtained by means of trial steps Fxi(x) = @F(x) = @xi F (x + ei) ; F (x) 2 + O( ) for all i =1(1)n Additional di culties arise here since for values of that are too small the subtraction is subject to rounding error, while for trial steps that are too large the neglect of terms O( 2 ) leads to incorrect values. The choice of suitable deviations requires special care in all cases (Hildebrand, 1956 Curtis and Reid, 1974). Cauchy (1847), Kantorovich (1940, 1945), Levenberg (1944), and Curry (1944) are the originators of the gradient strategy, which started life as a method of solving equations and systems of equations. It is rst referred to as an aid to solving variational problems by Hadamard (1908) and Courant (1943). Whereas Cauchy works with xed step lengths s (k) , Curry tries to determine the distance covered in the (not normalized) direction v (k) = ;rF (x (k) ) so as to reach a relative minimum (see also Brown, 1959). In principle, any one of the one dimensional search methods of Section 3.1 can be called upon to nd the optimal value for s (k) : F (x (k) + s (k) v (k) ) = min s fF (x (k) + sv (k) )g This variant of the basic strategy could thus be called a longest step procedure. It is better known however under the name optimum gradient method, ormethod ofsteepest descent (for maximization, ascent). Theoretical investigations of convergence and rate of convergence of the method can be found, e.g., in Akaike (1960), Goldstein (1962), Ostrowski (1967), Forsythe (1968), Elkin (1968), Zangwill (1969), and Wolfe (1969, 1970, 1971). Zangwill proves convergence based on the assumptions that the line searches are exact and the objective function is continuously twice di erentiable. Exactness of the one dimensional minimization is not, however, a necessary assumption (Wolfe, 1969). It is signi cant that one can only establish theoretically that a stationary point willbereached (rF (x) = 0) or approached (krF (x)k < "" > 0). The stationary point is a minimum, only if F (x) is convex and three times di erentiable (Akaike, 1960). Zellnik, Sondak, and Davis (1962), however, show that saddle points are in practice an obstacle, only if the search is started at one, or on a straight gradient trajectory passing through one. In other cases numerical rounding errors ensure that the path to a saddle point is unstable.
Multidimensional Strategies 67 The gradient strategy, however, cannot distinguish global from local minima. The optimum at which it aims depends only on the choice of the starting point for the search. The only chance of nding absolute extrema is to start su ciently often from various initial values of the variables and to iterate each timeuntil the convergence criterion is satis ed (Jacoby, Kowalik, and Pizzo, 1972). The termination rules usually recommended for gradient methods are that the absolute value of the vector or the di erence krF (x (k) )k
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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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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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