09 - GLOBAL MINIMIZATION
------------------------


      CALL EIUD09(N,XL,XU,NEXT,IERR)

        N - NUMBER OF VARIABLES (INPUT-DECLARED, OUTPUT-ACTUAL)
        XL(N) - LOWER BOUNDS (OUTPUT)
        XU(N) - UPPER BOUNDS (OUTPUT)
        NEXT - NO OF PROBLEM (FROM 1 TO 13)
        IERR - CONFLICT IN INPUT DATA IF IERR<>0 (OUTPUT)


      CALL EFFU09(N,X,F,NEXT)

        N - NUMBER OF VARIABLES (INPUT)
        X(N) - VECTOR OF VARIABLES (INPUT)
        F - VALUE OF THE MODEL FUNCTION (OUTPUT)
        NEXT - NUMBER OF PROBLEM (INPUT)


      CALL EFGU09(N,X,G,NEXT)

        N - NUMBER OF VARIABLES (INPUT)
        X(N) - VECTOR OF VARIABLES (INPUT)
        G(N) - GRADIENT OF THE OBJECTIVE FUNCTION (OUTPUT)
        NEXT - NUMBER OF PROBLEM (INPUT)


TEST FUNCTIONS FOR GLOBAL MINIMIZATION:
---------------------------------------

SPONA FUNCTION (NEXT=1)

      N=2

      -12 <= X(I) <= 10

      F=((X(1)-3.D0)*(X(1)+5.D0))**2+ &
         ((X(2)-2.D0)*(X(2)+3.D0))**2-(X(1)*X(2))**2

TRECCANI FUNCTION (NEXT=2)

      N=2

      -3 <= X(I) <= 3

      F=X(1)**4+4.0*X(1)**3+4.0*X(1)**2+X(2)**2

SIMPLIFIED ROSENBROCK'S FUNCTION (NEXT=3)

      N=2

      -3 <= X(I) <= 3

      F=0.5*(X(1)**2-X(2))**2 + (X(1)-1.0)**2

RASTRIGIN FUNCTION (NEXT=4)

      GLOBAL MINIMUM:
        NUMBER: 1
        FMIN: -2
        POINTS: (0,0)
        NUMBER LOC. MIN.: 50

      N=2

      -1 <= X(I) <= 1

      F=X(1)**2+X(2)**2-COS(18.0D0*X(1))-COS(18.0D0*X(2))

BRANIN FUNCTION (NEXT=5)
      GLOBAL MINIMUM
        NUMBER: 3
        FMIN: 0.399
        POINTS: (-33.142, 12.275),  (3.142, 2.275),  (9.425, 2.475)

      N=2

      -5 <= X(1) <= 10
       0 <= X(2) <= 15

      F=(X(2)-(5.1D0/(4.0D0*3.14D0**2))*X(1)**2+(5.0D0/3.14D0)*X(1)
     &  -6.0D0)**2+10.0D0*(1.0D0-(1.0D0/(8.0D0*3.140D0)))*COS(X(1))
     &  +10.0D0

GOLDSTEIN AND PRICE FUNCTION (NEXT=6)
     GLOBAL MINIMUM
       NUMBER: 1
       FMIN: 3
       POINTS: (0, -1)
       NUMBER LOC. MIN.: 4

      N=2

      -2 <= X(I) <= 2

      F=(1.0D0+(X(1)+X(2)+1.0D0)**2 * (19.0D0-14.0D0*X(1)
     &  +3.0D0*X(1)**2-14.0D0*X(2)+6.0D0*X(1)*X(2)+3.0D0*X(2)**2))
     &  *(30.0D0+((2.0D0*X(1)-3.0D0*X(2))**2) * (18.0D0-32.0D0*X(1)
     &  +12.0D0*X(1)**2+48.0D0*X(2)-36.0D0*X(1)*X(2)+27.0D0
     &  *X(2)**2))

3 HUMP CAMEL-BACK FUNCTION (NEXT=7)
      1 GLOBAL, 2 LOCAL MINIMA

      N=2

      -3 <= X(I) <= 3

      F=2.0*X(1)**2-1.05*X(1)**4+(X(1)**6)/6.0-X(1)*X(2)+X(2)**2

6 HUMP CAMEL-BACK FUNCTION (NEXT=8)

      N=2

      -3 <= X(I) <= 3

      F=4.0*X(1)**2-2.1*X(1)**4+(X(1)**6)/3.0-X(1)*X(2)
       &   -4.0*X(2)**2+4.0*X(2)**4

SHEKEL 5,7,9 FUNCTION (NEXT=9,10,11)
      LOCAL MINS ARE IN  ASH(I)

      N=4

      0 <= X(I) <= 10

      FF=0.0D0
      DO 20 I=1,M
        S=0.0D0
        DO 10 J=1,4
          S=S+(X(J)-ASH(J,I))**2
   10   CONTINUE
        F=F+1.0D0/(S+CSH(I))
   20 CONTINUE

HARTMAN 3 FUNCTION (NEXT=12)
      LOK. MINS ARE IN PHT(I)

      N=3

      0 <= X(I) <= 1

      F=0
      DO 20 I=1,4
        S=0
        DO 10 J=1,N
          S=S+AHT(I,J)*(X(J)-PHT(I,J))**2
   10   CONTINUE
        F=F+CHT(I)*EXP(-S)
   20 CONTINUE

HARTMAN 6 FUNCTION (NEXT=13)
      LOK. MINS ARE IN PHT(I)

      N=6

      0 <= X(I) <= 1

      F=0
      DO 20 I=1,4
        S=0
        DO 10 J=1,N
          S=S+AHT(I,J)*(X(J)-PHT(I,J))**2
   10   CONTINUE
        F=F+CHT(I)*EXP(-S)
   20 CONTINUE


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