# Try out NAG Library functions

Explore NAG maths and stats routines with interactive demos
Function ID
E04NCF
Name
nagf_opt_lsq_lincon_solve_old
Description
Solves linear least squares and convex quadratic programming problems (non-sparse)
Keywords
nonlinear least squares | data fitting | active-set | QP, quadratic programming | convex optimization | least squares
This example minimizes the function $\frac{1}{2}{‖b-Ax‖}^{2}$, where
 $A=111111111121111200113111-1-1-311141111111131111111211000-1111101111111011111110111223101111022 and b=1111111111$
subject to the bounds
 $0≤x1≤20≤x2≤2-∞≤x3≤20≤x4≤20≤x5≤20≤x6≤20≤x7≤20≤x8≤20≤x9≤2$
and to the general constraints
 $2.0≤x1+x2+x3+x4+x5+x6+x7+x8+4x9≤∞-∞≤x1+2x2+3x3+4x4-2x5+x6+x7+x8+x9≤2.01.0≤x1-x2+x3-x4+x5+x6+x7+x8+x9≤4.0$
The initial point, which is infeasible, is
 $x0=1.0,0.5,0.3333,0.25,0.2,0.1667,0.1428,0.125,0.1111T,$
and $F\left({x}_{0}\right)=9.4746$ (to five figures).
    Program e04ncfe

!     E04NCF Example Program Text

!     Mark 26.1 Release. NAG Copyright 2016.

!     .. Use Statements ..
Use nag_library, Only: dgemv, e04ncf, e04nef, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Parameters ..
Real (Kind=nag_wp), Parameter    :: one = 1.0_nag_wp
Real (Kind=nag_wp), Parameter    :: zero = 0.0_nag_wp
Integer, Parameter               :: inc1 = 1, nin = 5, nout = 6
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: obj
Integer                          :: i, ifail, iter, lda, ldc, liwork,    &
lwork, m, n, nclin, sdc
Logical                          :: verbose_output
!     .. Local Arrays ..
Real (Kind=nag_wp), Allocatable  :: a(:,:), b(:), bl(:), bu(:), c(:,:),  &
clamda(:), cvec(:), work(:), x(:)
Integer, Allocatable             :: istate(:), iwork(:), kx(:)
!     .. Intrinsic Procedures ..
Intrinsic                        :: max
!     .. Executable Statements ..
Write (nout,*) 'E04NCF Example Program Results'

!     Skip heading in data file

Read (nin,*) m, n, nclin
liwork = n
ldc = max(1,nclin)
lda = max(1,m)

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

!     This particular example problem is of type LS1, so we allocate
!     A(LDA,N), CVEC(1), B(M) and define LWORK as below

If (nclin>0) Then
lwork = 2*n**2 + 9*n + 6*nclin
Else
lwork = 9*n
End If

Allocate (istate(n+nclin),kx(n),iwork(liwork),c(ldc,sdc),bl(n+nclin),    &
bu(n+nclin),cvec(1),x(n),a(lda,n),b(m),clamda(n+nclin),work(lwork))

!     Set this to .True. to cause e04nqf to produce intermediate
!     progress output
verbose_output = .False.
If (.Not. verbose_output) Then
!       Switch off intermediate output from e04ncf
Call e04nef('Nolist')
Call e04nef('Print level = 0')
End If

!     Solve the problem

ifail = -1
Call e04ncf(m,n,nclin,ldc,lda,c,bl,bu,cvec,istate,kx,x,a,b,iter,obj,     &
clamda,iwork,liwork,work,lwork,ifail)

Select Case (ifail)
Case (0:5,7:)
!       Print variable headers
Write (nout,99999)

Do i = 1, n
Write (nout,99998) i, istate(i), x(i), clamda(i)
End Do

If (nclin>0) Then

!         C*x --> work
!         The NAG name equivalent of dgemv is f06paf
Call dgemv('N',nclin,n,one,c,ldc,x,inc1,zero,work,inc1)

!         Print constraint headers
Write (nout,99997)

Do i = 1, nclin
Write (nout,99996) i, istate(i+n), work(i), clamda(i+n)
End Do

End If

Write (nout,99995) obj
End Select

99999 Format (/,1X,'Varbl',3X,'Istate',4X,'Value',9X,'Lagr Mult')
99998 Format (1X,'V',2(1X,I3),4X,1P,E14.3,2X,1P,E12.3)
99997 Format (/,1X,'L Con',3X,'Istate',4X,'Value',9X,'Lagr Mult')
99996 Format (1X,'L',2(1X,I3),4X,1P,E14.3,2X,1P,E12.3)
99995 Format (/,1X,'Final objective value = ',1P,E15.3)
End Program e04ncfe

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