# Try out NAG Library functions

Explore NAG maths and stats routines with interactive demos
Function ID
E04USF
Name
nagf_opt_lsq_gencon_deriv_old
Description
Minimum of a sum of squares, nonlinear constraints, dense, active-set SQP method, using function values and optionally first derivatives
Keywords
nonlinear least squares | data fitting | quasi-Newton approximation | NLP | nonlinear programming | minimization, nonlinear constraints | sequential QP method | sum of squares
This example is based on Problem 57 in Hock and Schittkowski (1981) and involves the minimization of the sum of squares function
 $Fx=12∑i=144yi-fix2,$
where
 $fix=x1+0.49-x1e-x2ai-8$
and
 $iyiaiiyiai10.498230.412220.498240.402230.4810250.422440.4710260.402450.4810270.402460.4710280.412670.4612290.402680.4612300.412690.4512310.4128100.4312320.4028110.4514330.4030120.4314340.4030130.4314350.3830140.4416360.4132150.4316370.4032160.4316380.4034170.4618390.4136180.4518400.3836190.4220410.4038200.4220420.4038210.4320430.3940220.4122440.3942$
subject to the bounds
 $x1≥-0.4x2≥-4.0$
to the general linear constraint
 $x1+x2≥1.0$
and to the nonlinear constraint
 $0.49x2-x1x2≥0.09.$
The initial point, which is infeasible, is
 $x0=0.4,0.0T$
and $F\left({x}_{0}\right)=0.002241$.
!   E04USF Example Program Text
!   Mark 26.1 Release. NAG Copyright 2016.

Module e04usfe_mod

!     E04USF Example Program Module:
!            Parameters and User-defined Routines

!     .. Use Statements ..
Use nag_library, Only: nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Accessibility Statements ..
Private
Public                           :: confun, objfun
!     .. Parameters ..
Integer, Parameter, Public       :: nin = 5, nout = 6
Contains
Subroutine objfun(mode,m,n,ldfj,needfi,x,f,fjac,nstate,iuser,ruser)
!       Routine to evaluate the subfunctions and their 1st derivatives.

!       .. Parameters ..
Real (Kind=nag_wp), Parameter  :: a(44) = (/8.0E0_nag_wp,8.0E0_nag_wp, &
10.0E0_nag_wp,10.0E0_nag_wp,         &
10.0E0_nag_wp,10.0E0_nag_wp,         &
12.0E0_nag_wp,12.0E0_nag_wp,         &
12.0E0_nag_wp,12.0E0_nag_wp,         &
14.0E0_nag_wp,14.0E0_nag_wp,         &
14.0E0_nag_wp,16.0E0_nag_wp,         &
16.0E0_nag_wp,16.0E0_nag_wp,         &
18.0E0_nag_wp,18.0E0_nag_wp,         &
20.0E0_nag_wp,20.0E0_nag_wp,         &
20.0E0_nag_wp,22.0E0_nag_wp,         &
22.0E0_nag_wp,22.0E0_nag_wp,         &
24.0E0_nag_wp,24.0E0_nag_wp,         &
24.0E0_nag_wp,26.0E0_nag_wp,         &
26.0E0_nag_wp,26.0E0_nag_wp,         &
28.0E0_nag_wp,28.0E0_nag_wp,         &
30.0E0_nag_wp,30.0E0_nag_wp,         &
30.0E0_nag_wp,32.0E0_nag_wp,         &
32.0E0_nag_wp,34.0E0_nag_wp,         &
36.0E0_nag_wp,36.0E0_nag_wp,         &
38.0E0_nag_wp,38.0E0_nag_wp,         &
40.0E0_nag_wp,42.0E0_nag_wp/)
!       .. Scalar Arguments ..
Integer, Intent (In)           :: ldfj, m, n, needfi, nstate
Integer, Intent (Inout)        :: mode
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: f(m)
Real (Kind=nag_wp), Intent (Inout) :: fjac(ldfj,n), ruser(*)
Real (Kind=nag_wp), Intent (In) :: x(n)
Integer, Intent (Inout)        :: iuser(*)
!       .. Local Scalars ..
Real (Kind=nag_wp)             :: ai, temp, x1, x2
Integer                        :: i
Logical                        :: mode02, mode12
!       .. Intrinsic Procedures ..
Intrinsic                      :: exp
!       .. Executable Statements ..
x1 = x(1)
x2 = x(2)

If (mode==0 .And. needfi>0) Then
f(needfi) = x1 + (0.49E0_nag_wp-x1)*exp(-x2*(a(needfi)-8.0E0_nag_wp) &
)
Else

mode02 = (mode==0 .Or. mode==2)
mode12 = (mode==1 .Or. mode==2)

Do i = 1, m

ai = a(i)
temp = exp(-x2*(ai-8.0E0_nag_wp))

If (mode02) Then
f(i) = x1 + (0.49E0_nag_wp-x1)*temp
End If

If (mode12) Then
fjac(i,1) = 1.0E0_nag_wp - temp
fjac(i,2) = -(0.49E0_nag_wp-x1)*(ai-8.0E0_nag_wp)*temp
End If

End Do
End If

Return

End Subroutine objfun
Subroutine confun(mode,ncnln,n,ldcj,needc,x,c,cjac,nstate,iuser,ruser)
!       Routine to evaluate the nonlinear constraint and its 1st
!       derivatives.

!       .. Scalar Arguments ..
Integer, Intent (In)           :: ldcj, n, ncnln, nstate
Integer, Intent (Inout)        :: mode
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (Out) :: c(ncnln)
Real (Kind=nag_wp), Intent (Inout) :: cjac(ldcj,n), ruser(*)
Real (Kind=nag_wp), Intent (In) :: x(n)
Integer, Intent (Inout)        :: iuser(*)
Integer, Intent (In)           :: needc(ncnln)
!       .. Executable Statements ..
If (nstate==1) Then

!         First call to CONFUN.  Set all Jacobian elements to zero.
!         Note that this will only work when 'Derivative Level = 3'
!         (the default; see Section 11.2).

cjac(1:ncnln,1:n) = 0.0E0_nag_wp
End If

If (needc(1)>0) Then

If (mode==0 .Or. mode==2) Then
c(1) = -0.09E0_nag_wp - x(1)*x(2) + 0.49E0_nag_wp*x(2)
End If

If (mode==1 .Or. mode==2) Then
cjac(1,1) = -x(2)
cjac(1,2) = -x(1) + 0.49E0_nag_wp
End If

End If

Return

End Subroutine confun
End Module e04usfe_mod
Program e04usfe

!     E04USF Example Main Program

!     .. Use Statements ..
Use e04usfe_mod, Only: confun, nin, nout, objfun
Use nag_library, Only: e04usf, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: objf
Integer                          :: i, ifail, iter, lda, ldcj, ldfj,     &
ldr, liwork, lwork, m, n, nclin,     &
ncnln, sda, sdcjac
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), bl(:), bu(:), c(:),          &
cjac(:,:), clamda(:), f(:),          &
fjac(:,:), r(:,:), work(:), x(:),    &
y(:)
Real (Kind=nag_wp)               :: user(1)
Integer, Allocatable             :: istate(:), iwork(:)
Integer                          :: iuser(1)
!     .. Intrinsic Procedures ..
Intrinsic                        :: max
!     .. Executable Statements ..
Write (nout,*) 'E04USF Example Program Results'
Flush (nout)

!     Skip heading in data file

liwork = 3*n + nclin + 2*ncnln
lda = max(1,nclin)

If (nclin>0) Then
sda = n
Else
sda = 1
End If

ldcj = max(1,ncnln)

If (ncnln>0) Then
sdcjac = n
Else
sdcjac = 1
End If

ldfj = m
ldr = n

If (ncnln==0 .And. nclin>0) Then
lwork = 2*n**2 + 20*n + 11*nclin + m*(n+3)
Else If (ncnln>0 .And. nclin>=0) Then
lwork = 2*n**2 + n*nclin + 2*n*ncnln + 20*n + 11*nclin + 21*ncnln +    &
m*(n+3)
Else
lwork = 20*n + m*(n+3)
End If

Allocate (istate(n+nclin+ncnln),iwork(liwork),a(lda,sda),                &
bl(n+nclin+ncnln),bu(n+nclin+ncnln),y(m),c(max(1,                      &
ncnln)),cjac(ldcj,sdcjac),f(m),fjac(ldfj,n),clamda(n+nclin+ncnln),     &
r(ldr,n),x(n),work(lwork))

If (nclin>0) Then
End If