nlm
Non-Linear Minimization
This function carries out a minimization of the function f
using a Newton-type algorithm. See the references for details.
Usage
nlm(f, p, …, hessian = FALSE, typsize = rep(1, length(p)),
fscale = 1, print.level = 0, ndigit = 12, gradtol = 1e-6,
stepmax = max(1000 * sqrt(sum((p/typsize)^2)), 1000),
steptol = 1e-6, iterlim = 100, check.analyticals = TRUE)
Arguments
- f
the function to be minimized, returning a single numeric value. This should be a function with first argument a vector of the length of
p
followed by any other arguments specified by the…
argument.If the function value has an attribute called
gradient
or bothgradient
andhessian
attributes, these will be used in the calculation of updated parameter values. Otherwise, numerical derivatives are used.deriv
returns a function with suitablegradient
attribute and optionally ahessian
attribute.- p
starting parameter values for the minimization.
- …
additional arguments to be passed to
f
.- hessian
if
TRUE
, the hessian off
at the minimum is returned.- typsize
an estimate of the size of each parameter at the minimum.
- fscale
an estimate of the size of
f
at the minimum.- print.level
this argument determines the level of printing which is done during the minimization process. The default value of
0
means that no printing occurs, a value of1
means that initial and final details are printed and a value of 2 means that full tracing information is printed.- ndigit
the number of significant digits in the function
f
.- gradtol
a positive scalar giving the tolerance at which the scaled gradient is considered close enough to zero to terminate the algorithm. The scaled gradient is a measure of the relative change in
f
in each directionp[i]
divided by the relative change inp[i]
.- stepmax
a positive scalar which gives the maximum allowable scaled step length.
stepmax
is used to prevent steps which would cause the optimization function to overflow, to prevent the algorithm from leaving the area of interest in parameter space, or to detect divergence in the algorithm.stepmax
would be chosen small enough to prevent the first two of these occurrences, but should be larger than any anticipated reasonable step.- steptol
A positive scalar providing the minimum allowable relative step length.
- iterlim
a positive integer specifying the maximum number of iterations to be performed before the program is terminated.
- check.analyticals
a logical scalar specifying whether the analytic gradients and Hessians, if they are supplied, should be checked against numerical derivatives at the initial parameter values. This can help detect incorrectly formulated gradients or Hessians.
Details
Note that arguments after …
must be matched exactly.
If a gradient or hessian is supplied but evaluates to the wrong mode
or length, it will be ignored if check.analyticals = TRUE
(the
default) with a warning. The hessian is not even checked unless the
gradient is present and passes the sanity checks.
The C code for the “perturbed” cholesky, choldc()
has
had a bug in all R versions before 3.4.1.
From the three methods available in the original source, we always use method “1” which is line search.
The functions supplied should always return finite (including not
NA
and not NaN
) values: for the function value itself
non-finite values are replaced by the maximum positive value with a warning.
Value
A list containing the following components:
the value of the estimated minimum of f
.
the point at which the minimum value of
f
is obtained.
the gradient at the estimated minimum of f
.
the hessian at the estimated minimum of f
(if
requested).
an integer indicating why the optimization process terminated.
- 1:
relative gradient is close to zero, current iterate is probably solution.
- 2:
successive iterates within tolerance, current iterate is probably solution.
- 3:
last global step failed to locate a point lower than
estimate
. Eitherestimate
is an approximate local minimum of the function orsteptol
is too small.- 4:
iteration limit exceeded.
- 5:
maximum step size
stepmax
exceeded five consecutive times. Either the function is unbounded below, becomes asymptotic to a finite value from above in some direction orstepmax
is too small.
the number of iterations performed.
References
Dennis, J. E. and Schnabel, R. B. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ.
Schnabel, R. B., Koontz, J. E. and Weiss, B. E. (1985). A modular system of algorithms for unconstrained minimization. ACM Transactions on Mathematical Software, 11, 419--440. 10.1145/6187.6192.
See Also
constrOptim
for constrained optimization,
optimize
for one-dimensional
minimization and uniroot
for root finding.
deriv
to calculate analytical derivatives.
For nonlinear regression, nls
may be better.
Examples
library(stats)
# NOT RUN {
f <- function(x) sum((x-1:length(x))^2)
nlm(f, c(10,10))
nlm(f, c(10,10), print.level = 2)
utils::str(nlm(f, c(5), hessian = TRUE))
f <- function(x, a) sum((x-a)^2)
nlm(f, c(10,10), a = c(3,5))
f <- function(x, a)
{
res <- sum((x-a)^2)
attr(res, "gradient") <- 2*(x-a)
res
}
nlm(f, c(10,10), a = c(3,5))
## more examples, including the use of derivatives.
# }
# NOT RUN {
demo(nlm)
# }