An algorithm for general-purpose unconstrained non-linear optimization. The algorithm
is of quasi-Newton type with BFGS updating of the inverse
Hessian and soft line search with a trust region type monitoring of the
input to the line search algorithm. The interface of ‘ucminf’ is designed for
easy interchange with ‘optim’.
Usage
ucminf(par, fn, gr = NULL, ..., control = list(), hessian=0)
Arguments
par
Initial estimate of minimum for fn.
fn
Objective function to be minimized.
gr
Gradient of objective function. If NULL a finite difference
approximation is used.
...
Optional arguments passed to the objective and gradient functions.
control
A list of control parameters. See ‘Details’.
hessian
Integer value:
0
No hessian approximation is returned.
1
Returns a numerical approximation of the Hessian
using ‘hessian’ in the package ‘numDeriv’.
2
Returns final approximation of the inverse Hessian based on the series of BFGS
updates during optimization.
3
Same at 2, but will also return the Hessian (the inverse of 2).
If a TRUE or FALSE value is given it will switch
between option 1 or 0.
Details
The algorithm is documented in (Nielsen, 2000) (see References
below) together with a comparison to the Fortran subroutine
‘MINF’ and the Matlab function ‘fminunc’. The
implementation of ‘ucminf’ in R uses the
original Fortran version of the algorithm.
The interface in R is designed so that it is very easy to switch
between using ‘ucminf’ and ‘optim’. The
arguments par, fn, gr, and hessian
are all the same (with a few extra options for hessian in
‘ucminf’). The difference is that there is no method
argument in ‘ucminf’ and that some of the components in the
control argument are different due to differences in the
algorithms.
The algorithm can be given an initial estimate of the Hessian for the
optimization and it is possible to get the final approximation of the
Hessian based on the series of BFGS updates. This extra functionality
may be useful for optimization in a series of related problems.
The functions fn and gr can return Inf or NaN
if the functions cannot be evaluated at the supplied value, but the
functions must be computable at the initial value. The functions
are not allowed to return NA. Any names given to par will be
copied to the vectors passed to fn and gr. No
other attributes of par are copied over.
The control argument is a list that can supply any of the
following components:
trace
If trace is positive then detailed tracing information is printed for each iteration.
grtol
The algorithm stops when
||F'(x)||_inf <= grtol, that
is when the largest absolute value of the gradient is less than
grtol. Default value is grtol = 1e-6.
xtol
The algorithm stops when ||x-x_p||_2 <=
xtol*(xtol + ||x||_2), where x_p and x are the
previous and current estimate of the minimizer. Thus the algorithm
stops when the last relative step length is
sufficiently small. Default value is xtol = 1e-12.
stepmax
Initial maximal allowed step length (radius of
trust-region). The value is updated during the
optimization. Default value is stepmax = 1.
maxeval
The maximum number of function evaluations. A function evaluation
is counted as one evaluation of the objective function and of the
gradient function. Default value is maxeval = 500.
grad
Either ‘forward’ or ‘central’. Controls
the type of finite difference approximation to be used for the
gradient if no gradient function is given in the input argument
‘gr’. Default value is grad = 'forward'.
gradstep
Vector of length 2. The step length in finite
difference approximation for the gradient. Step length is
|x_i|*gradstep[1]+gradstep[2].
Default value is gradstep = c(1e-6, 1e-8).
invhessian.lt
A vector with an initial approximation to the lower triangle of the
inverse Hessian. If not given, the inverse Hessian is initialized
as the identity matrix. If H0 is the initial hessian matrix then
the lower triangle of the inverse of H0 can be found as
invhessian.lt = solve(H0)[lower.tri(H0,diag=TRUE)].
Value
par
Computed minimizer.
value
Objective function value at computed minimizer.
convergence
Flag for reason of termination:
1
Stopped by small gradient (grtol).
2
Stopped by small step (xtol).
3
Stopped by function evaluation limit (maxeval).
4
Stopped by zero step from line search
-2
Computation did not start: length(par) = 0.
-4
Computation did not start: stepmax is too small.
-5
Computation did not start: grtol or xtol <= 0.
-6
Computation did not start: maxeval <= 0.
-7
Computation did not start: given Hessian not pos. definite.
message
String with reason of termination.
hessian, invhessian
Estimate of (inv.) Hessian at computed minimizer. The type of
estimate is given by the input argument ‘hessian’.
invhessian.lt
The lower triangle of the final approximation to the
inverse Hessian based on the series of BFGS updates during optimization.
info
Information about the search:
maxgradient
||F'(x)||_inf, the
largest element in the absolute
value of the gradient at the computed minimizer.
laststep
Length of last step.
stepmax
Final maximal allowed step length.
neval
Number of calls to both objective and gradient function.
Author(s)
‘UCMINF’ algorithm design and Fortran code by Hans Bruun Nielsen.