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’.
ucminf(par, fn, gr = NULL, ..., control = list(), hessian = 0)ucminf returns a list of class 'ucminf'
containing the following elements:
Computed minimizer.
Objective function value at computed minimizer.
Flag for reason of termination:
Stopped by small gradient (grtol).
Stopped by small step (xtol).
Stopped by function evaluation limit (maxeval).
Stopped by zero step from line search
Computation did not start: length(par) = 0.
Computation did not start: stepmax is too small.
Computation did not start: grtol or xtol <= 0.
Computation did not start: maxeval <= 0.
Computation did not start: given Hessian not pos. definite.
String with reason of termination.
Estimate of (inv.) Hessian at computed minimizer. The type of estimate is given by the input argument ‘hessian’.
The lower triangle of the final approximation to the inverse Hessian based on the series of BFGS updates during optimization.
Information about the search:
\(||F'(x)||_\infty\), the largest element in the absolute value of the gradient at the computed minimizer.
Length of last step.
Final maximal allowed step length.
Number of calls to both objective and gradient function.
Initial estimate of minimum for fn.
Objective function to be minimized.
Gradient of objective function. If NULL a finite difference
approximation is used.
Optional arguments passed to the objective and gradient functions.
A list of control parameters. See ‘Details’.
Integer value:
Returns a numerical approximation of the Hessian using ‘hessian’ in the package ‘numDeriv’.
Returns final approximation of the inverse Hessian based on the series of BFGS updates during optimization.
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.
‘UCMINF’ algorithm design and Fortran code by Hans Bruun Nielsen.
K Hervé Dakpo took over maintenance of the package in May. 2023.
Implementation in R by Stig B. Mortensen, stigbm@gmail.com.
Modifications by Douglas Bates bates@stat.wisc.edu, Nov. 2010, to support nested optimization and correct issues with printing on Windows.
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:
grtolThe algorithm stops when
\(||F'(x)||_\infty \leq \) grtol, that is when the
largest absolute value of the gradient is less than grtol. Default value is
grtol = 1e-6.
xtolThe algorithm stops when \(||x-x_p||_2 \leq
\textrm{xtol}\cdot(\textrm{xtol} + ||x||_2)\), 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.
stepmaxInitial maximal allowed step length (radius of
trust-region). The value is updated during the optimization. Default value
is stepmax = 1.
maxevalThe 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.
gradEither ‘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'.
gradstepVector of length 2. The step length in finite
difference approximation for the gradient. Step length is
\(|x_i|\cdot\textrm{gradstep[1]+gradstep[2]}\). Default value is gradstep = c(1e-6, 1e-8).
invhessian.ltA 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)].
Nielsen, H. B. (2000) ‘UCMINF - An Algorithm For Unconstrained, Nonlinear Optimization’, Report IMM-REP-2000-19, Department of Mathematical Modelling, Technical University of Denmark. http://www.imm.dtu.dk/documents/ftp/tr00/tr19_00.pdf
The original Fortran source code was found at
http://www2.imm.dtu.dk/projects/hbn_software/ucminf.f.
(That URL is no longer available but archived at
https://web.archive.org/web/20050418082240/http://www.imm.dtu.dk/~hbn/Software/ucminf.f
-- Dr Nielsen passed away in 2015). The code has been slightly modified in
this package to be suitable for use with R.
The general structure of the implementation in R is based on the package ‘FortranCallsR’ by Diethelm Wuertz.
## Rosenbrock Banana function
fR <- function(x) (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2
gR <- function(x) c(-400 * x[1] * (x[2] - x[1] * x[1]) - 2 * (1 - x[1]),
200 * (x[2] - x[1] * x[1]))
# Find minimum and show trace
ucminf(par = c(2,.5), fn = fR, gr = gR, control = list(trace = 1))
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