Minimizes a function of N variables using a quasi-Newton method and a finite-difference gradient.
Required Arguments
FCN User-supplied subroutine to evaluate the function to be minimized. The usage is CALL FCN (N, X, F), where
N Length of X. (Input)
X The point at which
the function is evaluated. (Input)
X should not be
changed by FCN.
F The computed function value at the point X. (Output)
FCN must be declared EXTERNAL in the calling program.
X Vector of length N containing the computed solution. (Output)
Optional Arguments
N Dimension of
the problem. (Input)
Default: N = size
(X,1).
XGUESS Vector
of length N
containing an initial guess of the computed solution.
(Input)
Default: XGUESS = 0.0.
XSCALE Vector
of length N
containing the diagonal scaling matrix for the variables. (Input)
XSCALE is
used mainly in scaling the gradient and the distance between two points. In the
absence of other information, set all entries to 1.0.
Default: XSCALE = 1.0.
FSCALE Scalar
containing the function scaling. (Input)
FSCALE is used mainly
in scaling the gradient. In the absence of other information, set FSCALE to
1.0.
Default: FSCALE = 1.0.
IPARAM
Parameter vector of length 7. (Input/Output)
Set IPARAM(1) to zero for
default values of IPARAM and RPARAM. See
Comment 4.
Default: IPARAM = 0.
RPARAM
Parameter vector of length 7.(Input/Output)
See Comment 4.
FVALUE Scalar containing the value of the function at the computed solution. (Output)
FORTRAN 90 Interface
Generic: CALL UMINF (FCN, X [, ])
Specific: The specific interface names are S_UMINF and D_UMINF.
FORTRAN 77 Interface
Single: CALL UMINF (FCN, N, XGUESS, XSCALE, FSCALE, IPARAM, RPARAM, X, FVALUE)
Double: The double precision name is DUMINF.
Description
The routine UMINF uses a quasi-Newton method to find the minimum of a function f(x) of n variables. Only function values are required. The problem is stated as follows:
Given a starting point xc, the search direction is computed according to the formula
d = -B-1 gc
where B is a positive definite approximation of the Hessian and gc is the gradient evaluated at xc. A line search is then used to find a new point
xn = xc + λd, λ > 0
such that
f(xn) ≤ f(xc) + αgT d, α ∈ (0, 0.5)
Finally, the optimality condition ||g(x)|| = ɛ is checked where ɛ is a gradient tolerance.
When optimality is not achieved, B is updated according to the BFGS formula
where s = xn - xc and y = gn - gc. Another search direction is then computed to begin the next iteration. For more details, see Dennis and Schnabel (1983, Appendix A).
Since a finite-difference method is used to estimate the gradient, for some single precision calculations, an inaccurate estimate of the gradient may cause the algorithm to terminate at a noncritical point. In such cases, high precision arithmetic is recommended. Also, whenever the exact gradient can be easily provided, IMSL routine UMING should be used instead.
Comments
1. Workspace may be explicitly provided, if desired, by use of U2INF/DU2INF. The reference is:
CALL U2INF (FCN, N, XGUESS, XSCALE, FSCALE, IPARAM, RPARAM, X, FVALUE, WK)
The additional argument is:
WK Work vector of length N(N + 8). WK contains the following information on output: The second N locations contain the last step taken. The third N locations contain the last Newton step. The fourth N locations contain an estimate of the gradient at the solution. The final N2 locations contain the Cholesky factorization of a BFGS approximation to the Hessian at the solution.
2. Informational errors
Type Code
4 2 The iterates appear to be converging to a noncritical point.
4 3 Maximum number of iterations exceeded.
4 4 Maximum number of function evaluations exceeded.
4 5 Maximum number of gradient evaluations exceeded.
4 6 Five consecutive steps have been taken with the maximum step length.
2 7 Scaled step tolerance satisfied; the current point may be an approximate local solution, or the algorithm is making very slow progress and is not near a solution, or STEPTL is too big.
3 8 The last global step failed to locate a lower point than the current X value.
3. The first stopping criterion for UMINF occurs when the infinity norm of the scaled gradient is less than the given gradient tolerance (RPARAM(1)). The second stopping criterion for UMINF occurs when the scaled distance between the last two steps is less than the step tolerance (RPARAM(2)).
4. If the default parameters are desired for UMINF, then set IPARAM(1) to zero and call the routine UMINF. Otherwise, if any nondefault parameters are desired for IPARAM or RPARAM, then the following steps should be taken before calling UMINF:
CALL U4INF (IPARAM, RPARAM)
Set nondefault values for
desired IPARAM, RPARAM elements.
Note that the call to U4INF will set IPARAM and RPARAM to their default values so only nondefault values need to be set above.
The following is a list of the parameters and the default values:
IPARAM Integer vector of length 7.
IPARAM(1) = Initialization flag.
IPARAM(2) = Number of good digits in the function.
Default: Machine dependent.
IPARAM(3) = Maximum number of iterations.
Default: 100.
IPARAM(4) = Maximum number of function evaluations.
Default: 400.
IPARAM(5) = Maximum number of gradient evaluations.
Default: 400.
IPARAM(6) = Hessian initialization
parameter.
If IPARAM(6) = 0, the Hessian is initialized
to the identity matrix; otherwise, it is initialized to a diagonal matrix
containing
on the diagonal where t = XGUESS, fs = FSCALE, and s = XSCALE.
Default: 0.
IPARAM(7) = Maximum number of Hessian evaluations.
Default: Not used in UMINF.
RPARAM Real vector of length 7.
RPARAM(1) = Scaled gradient
tolerance.
The i-th component of the scaled gradient at
x is
calculated as
where g = ∇f (x), s = XSCALE, and fs = FSCALE.
Default:
in double where ɛ is the machine precision.
RPARAM(2) = Scaled step tolerance. (STEPTL)
The
i-th component of the scaled step between two points x and
y is computed as
where s = XSCALE.
Default: ɛ2/3 where ɛ is the machine precision.
RPARAM(3) = Relative function tolerance.
Default: Not used in UMINF.
RPARAM(4) = Absolute function tolerance.
Default: Not used in UMINF.
RPARAM(5) = False convergence tolerance.
Default: Not used in UMINF.
RPARAM(6) = Maximum allowable step size.
Default: 1000 max(ɛ1, ɛ2) where
RPARAM(7) = Size of initial trust region radius.
Default: Not used in UMINF.
If double precision is required, then DU4INF is called, and RPARAM is declared double precision.
5. Users wishing to override the default print/stop attributes associated with error messages issued by this routine are referred to Error Handling in the Introduction.
Example
The function
is minimized.
USE UMINF_INT
USE U4INF_INT
USE UMACH_INT
IMPLICIT NONE
INTEGER N
PARAMETER (N=2)
!
INTEGER IPARAM(7), L, NOUT
REAL F, RPARAM(7), X(N), XGUESS(N), &
XSCALE(N)
EXTERNAL ROSBRK
!
DATA XGUESS/-1.2E0, 1.0E0/
!
! Relax gradient tolerance stopping
! criterion
RPARAM(1) = 10.0E0*RPARAM(1)
! Minimize Rosenbrock function using
! initial guesses of -1.2 and 1.0
CALL UMINF (ROSBRK, X,
XGUESS=XGUESS, IPARAM=IPARAM, RPARAM=RPARAM,
&
FVALUE=F)
! Print results
CALL UMACH (2, NOUT)
WRITE (NOUT,99999) X, F, (IPARAM(L),L=3,5)
!
99999 FORMAT (' The solution is ', 6X, 2F8.3, //, ' The function ', &
'value is ', F8.3, //, ' The number of iterations is ', &
10X, I3, /, ' The number of function evaluations is ', &
I3, /, ' The number of gradient evaluations is ', I3)
!
END
!
SUBROUTINE ROSBRK (N, X, F)
INTEGER N
REAL X(N), F
!
F = 1.0E2*(X(2)-X(1)*X(1))**2 + (1.0E0-X(1))**2
!
RETURN
END
Output
The solution
is 1.000
1.000
The function value is 0.000
The number of
iterations is
15
The number of function evaluations is 40
The number of gradient
evaluations is 19
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