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and without building local approximation models (in contrast to many derivate-free optimizers). The following figure illustrates an example of 2-dimensional
Rosenbrock function optimization by
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Many of the stationary points of the function exhibit a regular pattern when plotted. This structure can be exploited to locate them.
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944:{\displaystyle f(\mathbf {x} )=\sum _{i=1}^{N-1}\quad {\mbox{where}}\quad \mathbf {x} =(x_{1},\ldots ,x_{N})\in \mathbb {R} ^{N}.}
1658:
Kok, Schalk; Sandrock, Carl (2009). "Locating and
Characterizing the Stationary Points of the Extended Rosenbrock Function".
1588:
1164:. This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a
1103:
152:
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The
Rosenbrock function can be efficiently optimized by adapting appropriate coordinate system without using any
117:
143:-shaped flat valley. To find the valley is trivial. To converge to the global minimum, however, is difficult.
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after 185 function evaluations. The figure below visualizes the evolution of the algorithm.
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716:{\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\dots ,x_{N})=\sum _{i=1}^{N/2}\left.}
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1540:"An automatic method for finding the greatest or least value of a function"
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Function used as a performance test problem for optimization algorithms
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with a regular initial simplex a minimum is found with function value
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this method breaks down due to the size of the coefficients involved.
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19:
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Computer Aided
Graphing and Simulation Tools for AutoCAD users
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the function is symmetric and the minimum is at the origin.
470:
uncoupled 2D Rosenbrock problems, and is defined only for
1606:"Effect of Rounding Errors on the Variable Metric Method"
438:
Animation of
Rosenbrock's function of three variables.
23:
Plot of the
Rosenbrock function of two variables. Here
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Nelder-Mead method applied to the
Rosenbrock function
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1610:Journal of Optimization Theory and Applications
726:This variant has predictably simple solutions.
237:{\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}}
345:. Usually, these parameters are set such that
1418:can be found after 325 function evaluations.
1657:
1583:(1st ed.). Boca Raton, FL: CRC Press.
1305:Rosenbrock roots exhibiting hump structures
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431:Two variants are commonly encountered.
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1388:. The solution with the function value
1216:can be used to determine the number of
1733:
1648:
1604:Dixon, L. C. W.; Mills, D. J. (1994).
61:, and the minimum value of zero is at
1712:
1293:
1636:"Generalized Rosenbrock's function"
729:A second, more involved variant is
13:
1502:{\displaystyle 1.36\cdot 10^{-10}}
14:
1762:
1700:
397:. Only in the trivial case where
1111:
881:
746:
510:
427:Multidimensional generalizations
1741:Test functions for optimization
1519:Test functions for optimization
879:
871:
297:{\displaystyle (x,y)=(a,a^{2})}
1707:Rosenbrock function plot in 3D
1628:
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1572:
1531:
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1375:
1357:
1263:{\displaystyle |x_{i}|<2.4}
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1212:can be determined exactly and
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1381:{\displaystyle x_{0}=(-3,-4)}
1043:{\displaystyle 4\leq N\leq 7}
1018:) and exactly two minima for
1466:{\displaystyle x_{0}=(-1,1)}
954:has exactly one minimum for
130:Rosenbrock's banana function
116:in 1960, which is used as a
7:
1512:
1335:adaptive coordinate descent
1093:{\displaystyle (1,1,...,1)}
247:It has a global minimum at
146:The function is defined by
10:
1767:
1672:10.1162/evco.2009.17.3.437
139:is inside a long, narrow,
1579:Simionescu, P.A. (2014).
1538:Rosenbrock, H.H. (1960).
1223:, while the roots can be
1100:and a local minimum near
99:mathematical optimization
54:{\displaystyle a=1,b=100}
1660:Evolutionary Computation
1411:{\displaystyle 10^{-10}}
338:{\displaystyle f(x,y)=0}
118:performance test problem
1050:—the global minimum at
1011:{\displaystyle (1,1,1)}
1751:Functions and mappings
1557:10.1093/comjnl/3.3.175
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124:. It is also known as
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1717:"Rosenbrock Function"
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1310:Optimization examples
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390:{\displaystyle b=100}
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86:{\displaystyle (1,1)}
56:
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1544:The Computer Journal
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1425:from starting point
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1337:from starting point
1331:gradient information
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114:Howard H. Rosenbrock
65:
27:
973:{\displaystyle N=3}
825:
637:
463:{\displaystyle N/2}
416:{\displaystyle a=0}
364:{\displaystyle a=1}
126:Rosenbrock's valley
103:Rosenbrock function
1714:Weisstein, Eric W.
1622:10.1007/BF02196600
1499:
1463:
1423:Nelder–Mead method
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1590:978-1-4822-5290-3
1294:Stationary points
1283:{\displaystyle N}
1227:in the region of
1201:{\displaystyle N}
1181:{\displaystyle x}
1166:rational function
1118:
876:
486:{\displaystyle N}
120:for optimization
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1701:External links
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1550:(3): 175–184.
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1270:. For larger
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137:global minimum
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1666:(3): 437–53.
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1188:. For small
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1639:. Retrieved
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1746:Polynomials
1616:: 175–179.
1210:polynomials
1735:Categories
1681:2263/13845
1641:2008-09-16
1525:References
1421:Using the
122:algorithms
1722:MathWorld
1566:0010-4620
1492:−
1484:⋅
1449:−
1401:−
1370:−
1361:−
1143:…
1128:−
1116:^
1035:≤
1029:≤
924:∈
905:…
846:−
809:−
776:−
758:∑
690:−
682:−
639:−
626:−
576:∑
553:…
212:−
184:−
141:parabolic
105:is a non-
1690:19708775
1513:See also
304:, where
110:function
1225:bounded
1688:
1587:
1564:
107:convex
101:, the
1221:roots
875:where
1686:PMID
1585:ISBN
1562:ISSN
1481:1.36
1255:<
1218:real
1208:the
980:(at
472:even
371:and
135:The
1676:hdl
1668:doi
1618:doi
1552:doi
1258:2.4
1168:of
787:100
609:100
493:s:
385:100
128:or
97:In
49:100
1737::
1719:.
1684:.
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1664:17
1662:.
1650:^
1614:80
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1554::
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1461:)
1458:1
1455:,
1452:1
1446:(
1443:=
1438:0
1434:x
1376:)
1373:4
1367:,
1364:3
1358:(
1355:=
1350:0
1346:x
1278:N
1251:|
1245:i
1241:x
1236:|
1196:N
1176:x
1152:)
1149:1
1146:,
1140:,
1137:1
1134:,
1131:1
1125:(
1122:=
1112:x
1088:)
1085:1
1082:,
1079:.
1076:.
1073:.
1070:,
1067:1
1064:,
1061:1
1058:(
1038:7
1032:N
1026:4
1006:)
1003:1
1000:,
997:1
994:,
991:1
988:(
968:3
965:=
962:N
939:.
934:N
929:R
921:)
916:N
912:x
908:,
902:,
897:1
893:x
889:(
886:=
882:x
869:]
864:2
860:)
854:i
850:x
843:1
840:(
837:+
832:2
828:)
822:2
817:i
813:x
804:1
801:+
798:i
794:x
790:(
784:[
779:1
773:N
768:1
765:=
762:i
754:=
751:)
747:x
743:(
740:f
711:.
707:]
701:2
697:)
693:1
685:1
679:i
676:2
672:x
668:(
665:+
660:2
656:)
650:i
647:2
643:x
634:2
629:1
623:i
620:2
616:x
612:(
605:[
599:2
595:/
591:N
586:1
583:=
580:i
572:=
569:)
564:N
560:x
556:,
550:,
545:2
541:x
537:,
532:1
528:x
524:(
521:f
518:=
515:)
511:x
507:(
504:f
481:N
458:2
454:/
450:N
411:0
408:=
405:a
382:=
379:b
359:1
356:=
353:a
333:0
330:=
327:)
324:y
321:,
318:x
315:(
312:f
292:)
287:2
283:a
279:,
276:a
273:(
270:=
267:)
264:y
261:,
258:x
255:(
230:2
226:)
220:2
216:x
209:y
206:(
203:b
200:+
195:2
191:)
187:x
181:a
178:(
175:=
172:)
169:y
166:,
163:x
160:(
157:f
93:.
81:)
78:1
75:,
72:1
69:(
46:=
43:b
40:,
37:1
34:=
31:a
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