Evolution and Optimum Seeking

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166 Comparison of Direct Search Strategies for Parameter Optimization was best from the point of view of its results. So long as no generally recognized quality function of this kind exists, the question of which optimization method is optimal remains unanswered. 6.2 Theoretical Results Classical optimization theory is concerned with establishing necessary and su cient existence criteria for maxima and minima. It provides systems of equations but no iterative methods of nding their solutions. Not even Dantzig's simplex method (1966) for solving linear programming problems can be regarded as a direct result of theory{theoretical considerations of the linear problem only show that the extremum sought, except in special cases, must always lie in a corner of the polyhedron de ned by the constraints. With n variables and m constraints (together with n non-negativity conditions) the number of corners or points of intersection of the hypersurfaces formed by the constraints is also limited to a maximum of m+n . Even the systematic inspection of all the points of n intersection would be a nite optimization method. But not all the points of intersection are also within the allowed region (Saaty, 1955, 1963). Muller-Merbach (1971) gives mn ; m + 2 as an upper bound to the number of feasible corner points. The simplex method, which is a method of steepest ascent along the edges of the polyhedron only traverses a tiny fraction of all the corners. Dantzig (1966) refers to empirical evidence that the number of necessary iterations increases as n, thenumber of variables, if the number of constraints m is constant, or as m if (n ; m) is not too small. Since, in the least favorable case, between m and 2 m exchange operations must be performed on the tableau of (m +1)(n + 1) coe cients, the average computation time increases as O(m2 n). In so-called degenerate cases, however, the simplex method can also become in nite. The repeated cycling through the same corners must then be broken by arule for randomly choosing the iteration step (Dantzig). From a theoretical point of view the ellipsoid method of Khachiyan (1979) and the interior point method of Karmarkar (1984) do have the advantage of polynomial time consumption even in the worst case. The question of niteness of iterative methodsisalsoacentral theme of non-linear programming. In this case the solution can lie at any point on the border or interior of the enclosed region. For the special case that the objective function and all the constraint functions are convex and multiply di erentiable Kuhn and Tucker (1951) and John (1948) have derived necessary and su cient conditions for extremal solutions. Most of the iteration methods that have beendeveloped on this basis are designed for problems with a quadratic objective function and linear constraints. Representative of quadratic programming are, for example, the methods of Beale (1956) and Wolfe (1959a). They make extensive use of the algorithm of the simplex method and thus belong, according to Hadley (1969), to the class of neighboring extremal point methods. Other strategies can moveinto the allowed region in the course of the iterations. As far as the constraints permit they take the direction of the gradient of the objective function. They are therefore known as gradient methods of non-linear programming (Kappler, 1967). As their name may suggest, however, they are not suitable for all non-linear problems. Their convergence can be proved at best for di erentiable quasi-convex programs (Kunzi, Krelle, and
Theoretical Results 167 Oettli, 1962). For these conditions the number of required iterations and rate of convergence cannot be stated in general. The same is true for the methods of Khachiyan (1979) and Karmarkar (1984). In the following chapters a short summary is attempted of the convergence properties of non-linear optimization methods in the unconstrained case (hill climbing methods). 6.2.1 Proofs of Convergence A proof of convergence of an iterative method will aim to show that a sequence of iteration points x (k) tends monotonically with the index k towards the point x 0 which is sought: or lim k!1 kx(k) ; x 0 k!0 kx (k) ; x 0 k " " 0 for 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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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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Page 123 and 124: The Two Membered Evolution Strategy
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Page 165 and 166: Genetic Algorithms 157 out any chan
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Page 169 and 170: Simulated Annealing 161 There are t
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Page 173: Chapter 6 Comparison of Direct Sear
Page 177 and 178: Theoretical Results 169 where 0
Page 179 and 180: Theoretical Results 171 tions. Simi
Page 181 and 182: 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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