p.specht
 | Das Verfahren de Nelder & Mead es una relativ einfache, por lo tanto robuste Búsqueda después de Minima o. Maxima en uno zwei- oder mehrdimensionalen Fehler-, Kosten- oder Gewinnfunktion (Im Gegensatz a otro Verfahren kommt lo sin partielle Ableitungen de!).
Como todos solche Verfahren besteht auch hier el Gefahr, en bloß lokalen Optimalpunkten hängenzubleiben. Daher debería uno stets mehrere Duchläufe con geänderten Anfangs-Eckpunkten des "wandernden Dreiecks" (2D) o. wandernden Tetraeders (3D) o. Simplexes (>3D) durchspielen.
Der Algorithmus incluso stammt de el Jahre 1965 y fue de HP-BASIC después de XProfan 11 transkribiert. Daher todavía el alte "Spagetti-Programmierung" con vielen "GOTO". Versuche verliefen en me sin große Fehler. Hier mi Umsetzung después de XProfan 11 - aber como siempre sin jede Gewähr:
Título de la ventana "Nelder-Mead Downhillsimplex a Optimierung en n-dimensional-nichtlinearen Funktionen"
' (Tr) 2012-07 traducido de HP-Basic después de XProfan 11.2a by P. Pájaro carpintero, Wien
' Umsetzung sin jegliche Gewähr!
REM Rechtsvermerke el Originalvorlage:
REM Copyright 1987
REM John H. Mathews
REM Dept. of Mathematics
REM California State University, Fullerton
REM Nelder-Mead Method for N-Dimensions
Ventana 0,0 - %maxx,%maxy-44
CLS:Font 2
Declarar N&,K&,J&,FUN$,L$,R$,A$,ANS$
Declarar Epsilon!,x!,y!,z!,u!,v!,w!,F!,S!,Norm!,YR!,YE!,YC!
Declarar LO&,HO&,HI&,LI&,Max&,Min&,Count&
Goto "L100"
Proc S10' FUNCTION
X! = P![1] : Y! = P![2] : Z! = P![3]
U! = P![4] : V! = P![5] : W!=P![6]
'====================================
F! = X!*X! - 4*X! + Y!*Y! - Y! - X!*Y!
'====================================
'Weitere Testfunktionen:
' Rosenbrock-„Bananenfunktion“
'f(x,y)=(1-x)^2+100*(y-x^2)^2
'min en 1,1 con f(x,y)=0
' Himmelblau-Función
'f(x,y)=(x^2+y-11)^2+(x+y^2-7)^2
'max en x=-0.270845, y=-0.923039 con f(x,y)=181.617
'4*min: f(3.0,2.0)=0.0 f(-2.8051118,3.131312)=0.0
'f(-3.779310,-3.283186)=0.0 f(3.584428, -1.848126)=0.0
'====================================
ENDPROC
' Proc PRINT
Proc S20
N&=2
FUN$ ="X*X - 4*X + Y*Y - Y - X*Y"
C$[1] = "[X' : C$[2] = ",Y" : C$[3] = ",Z"
C$[4] = ",U" : C$[5] = ",V" : C$[6] = ",W,"
L$ =" F" : R$ ="] = "
Imprimir L$;C$[1];
WhileLoop 2,N&: K&=&Loop
Imprimir C$[K&];
EndWhile
Imprimir R$;FUN$
Volver
ENDPROC
'{ Hauptteil PROGRAM NELDER MEAD
L100:
Epsilon! = +1*10^-5
Declarar C$[6],C![6],E![6],M![6],P![6],R![6],V![6,6],Y![6],Z![6]
L120:
REM SUBROUTINE INPUTS
S1000
REM SUBROUTINE NELDER MEAD
Gosub "S200"
REM SUBROUTINE OUTPUT
S2000
Imprimir "\n WANT TO TRY NEW STARTING VERTICES ? ";
Entrada ANS$
Case (ANS$ = "Y") Or (ANS$ = "y") : Goto "L120"
Goto "L9999"
'}
S200:
REM SUBROUTINE NELDER MEAD
MIN&=10
MAX&=200
COUNT&=0
REM Order the vertices.
LO&=0 : HI& =0
WhileLoop N& : J&=&Loop
caso Y![J&] < Y![LO&] : LO&=J&
caso Y![J&] > Y![HI&] : HI&=J&
EndWhile
LI&=HI& : HO&=LO&
WhileLoop 0,n&:J&=&Loop
caso (J& <> LO&) AND (Y![J&] < Y![LI&]) : LI&=J&
caso (J& <> HI&) AND (Y![J&] > Y![HO&]): HO&=J&
EndWhile
' =======================================================================
Mientras que (Y![HI&] > (Y![LO&] + EPSILON!)) And ( (COUNT&<MAX&) Or (COUNT&<MIN&) )
REM The main bucle
REM Form new vertices M and R
WhileLoop N&:K&=&Loop
S!=0
WhileLoop 0,N& : J& = &Loop
S!=S!+V![J&,K&]
EndWhile
M![K&]=(S!-V![HI&,K&]) / N&
EndWhile
WhileLoop N&: K&=&Loop
R![K&]=2*M![K&]-V![HI&,K&]
P![K&]=R![K&]
EndWhile
REM SUBROUTINE FUNCTION
S10
YR!=F!
REM Improve the simplex.
If YR! < Y![HO&] : Goto "L375" : Más : Goto "L525" : EndIf
L375:
IF Y![LI&] < YR! : Goto "L380" : Más : Goto "L410" : EndIf
L380:
REM Replace a vertex
WhileLoop N&:K&=&Loop
V![HI&,K&]=R![K&]
EndWhile
Y![HI&]=YR!
Goto "L515"
L410:
REM Construct vertex E.
WhileLoop N&: K&=&Loop
E![K&]=2*R![K&]-M![K&]
P![K&]=E![K&]
EndWhile
REM DO SUBROUTINE FUNCTION
S10
YE!=F!
IF YE! < Y![LI&] : GOTO "L455" : ELSE : GOTO "L480" : ENDIF
L455:
WhileLoop N&: K&=&Loop
V![HI&,K&]=E![K&]
EndWhile
Y![HI&]=YE!
Goto "L510"
REM ELSE
L480:
REM Replace a vertex.
WhileLoop N&: K&=&Loop
V![HI&,K&]=R![K&]
EndWhile
Y![HI&]=YR!
L510:
REM ENDIF
L515:
REM ENDIF
Goto "L700"
L525:
If YR! < Y![HI&] : Goto "L535" : Más : Goto "L560" : EndIf
L535:
REM Replace a vertex
WhileLoop N& : K&=&Loop
V![HI&,K&]=R![K&]
EndWhile
Y![HI&]=YR!
L560:
REM CONTINUE
REM Construct vertex C
WhileLoop N& : K& = &Loop
C![K&]=(V![HI&,K&]+M![K&])/2
P![K&]=C![K&]
EndWhile
S10
' REM SUBROUTINE FUNCTION
YC!=F!
If YC! < Y![HI&] : Goto "L605" : Más : Goto "L630" : EndIf
L605:
WhileLoop N&: K&=&Loop
V![HI&,K&]=C![K&]
EndWhile
Y![HI&]=YC!
Goto "L695"
L630:
REM ELSE
REM Shrink the simplex
WhileLoop 0,N& : J&=&Loop
If J& <> LO& : Goto "L650" : Más : Goto "L685" : EndIf
L650:
WhileLoop N& : K& = &Loop
V![J&,K&]=(V![J&,K&]+V![LO&,K&])/2
Z![K&]=V![J&,K&]
P![K&] = Z![K&]
EndWhile
S10
' REM SUBROUTINE FUNCTION
Y![J&]=F!
L685:
REM CONTINUE
EndWhile
L695:
REM ENDIF
L700:
REM ENDIF
COUNT&=COUNT&+1
REM Order the vertices.
LO&=0
HI&=0
WhileLoop N& : J&=&Loop
caso Y![J&] < Y![LO&] : LO&=J&
caso Y![J&] > Y![HI&] : HI& =J&
EndWhile
LI&=HI&
HO&=LO&
WhileLoop 0,N&:J&=&Loop
caso (J& <> LO&) AND (Y![J&] < Y![LI&]) : LI&=J&
caso (J& <> HI&) AND (Y![J&] > Y![HO&]): HO&=J&
EndWhile
WEND
' =======================================================================
REM Determine the size of the simplex
NORM!=0
WhileLoop 0,N&:J&=&Loop
S!=0
WhileLoop N&: K&=&Loop
S!=S! + ( V![LO&,K&] - V![J&,K&] ) * ( V![LO&,K&] - V![J&,K&] )
EndWhile
caso S! > NORM! : NORM!=S!
EndWhile
NORM!=SQRT(NORM!)
Volver
REM SUBROUTINE INPUTS
Proc S1000
CLS
Imprimir " THE NELDER-MEAD SIMPLEX METHOD O 'POLYTOPE METHOD' IS"
Imprimir "\n USED FOR FINDING THE MINIMUM OF THE FUNCTION F(V)"
Imprimir "\n FOR CONVENIENCE, THE FUNCTION F(V) CAN BE EXPRESSED USING"
Imprimir "\n THE VARIABLES X = v , Y = v , Z = v , U = v , V = v , W = v ."
Imprimir " 1 2 3 4 5 6 "
Imprimir
' REM DO SUBROUTINE PRINT FUNCTION
S20
Imprimir "\n YOU MUST SUPPLY",int(N&+1)," LINEARLY INDEPENDENT"
Imprimir
If N& = 2 : Goto "L1160" : Más : Goto "L1190" : EndIf
L1160:
Imprimir " STARTING POINTS V = ( v , v ) FOR J=0,1,3"
Imprimir " J J,1 J,2"
Goto "L1280"
L1190:
REM ELSE
If N& = 3 : Goto "L1210" : Más : Goto "L1240" : EndIf
L1210:
Imprimir " STARTING POINTS V = (v,v,v) FOR J=0,1,3,4"
Imprimir " J J,1 J,2 J,3"
Goto "L1280"
L1240:
REM ELSE
PRINT " STARTING POINTS V = (v,v,...,v) FOR J=0,1,...,N"
PRINT " J J,1 J,2 J,N"
PRINT " WHERE N =",N&
L1280:
REM ENDIF
WhileLoop 0,N& : J&=&Loop
PRINT
PRINT " GIVE COORDINATES OF POINT V"
PRINT " ",J&
WhileLoop N& : K&=&Loop
PRINT " V(";J&;",";K&;") = ";
INPUT A$
V![J&,K&]=val(A$)
Z![K&]=V![J&,K&]
EndWhile
WhileLoop N& : K&=&Loop
P![K&] = Z![K&]
EndWhile
' REM DO SUBROUTINE FUNCTION
S10
Y![J&]=F!
EndWhile' NEXT J&
Volver
ENDPROC
' SUBROUTINE OUTPUT
Proc S2000
CLS
Imprimir
Imprimir " THE NELDER-MEAD METHOD WAS USED TO FIND THE MINIMUM OF THE FUNCTION"
Imprimir
' REM DO SUBROUTINE PRINT FUNCTION
S20
Imprimir
Imprimir " IT TOOK ";COUNT&;" ITERATIONS TO FIND AN APPROXIMATION FOR"
Imprimir
IF N& = 2 : Goto "L2100" : Más : Goto "L2130" : EndIf
L2100:
Imprimir " THE COORDINATES OF THE LOCAL MINIMUM P = (p ,p )"
Imprimir " 1 2"
Goto "L2220"
L2130:
REM ELSE
If N& = 3 : Goto "L2150" : Más : Goto "L2180" : EndIf
L2150:
Imprimir " THE COORDINATES OF THE LOCAL MINIMUM P = (p ,p ,p )"
Imprimir " 1 2 3"
Goto "L2220"
L2180:
REM ELSE
Imprimir " THE COORDINATES OF THE LOCAL MINIMUM P = (p ,p ,...,p )"
Imprimir " 1 2 N"
Imprimir " WHERE N = ";N&
L2220:
REM ENDIF
WhileLoop N& : K&=&Loop
Imprimir "P(";K&;") = ";V![LO&,K&]
EndWhile
Imprimir "\n THE MAXIMUM DISTANCE TO THE OTHER VERTICES OF THE SIMPLEX IS"
Imprimir "\n DP = ";Norm!
Imprimir "\n THE FUNCTION VALUE AT THE MINIMUM POINT IS"
Imprimir "\n F(P) = ";Y![LO&]
Imprimir "\n DF = ";Y![HI&]-Y![LO&];" IS AN ESTIMATE FOR THE ACCURACY."
Volver
ENDPROC
L9999:
End
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