optimize
One Dimensional Optimization
The function optimize
searches the interval from
lower
to upper
for a minimum or maximum of
the function f
with respect to its first argument.
optimise
is an alias for optimize
.
 Keywords
 optimize
Usage
optimize(f = , interval = , ..., lower = min(interval), upper = max(interval), maximum = FALSE, tol = .Machine$double.eps^0.25)
optimise(f = , interval = , ..., lower = min(interval), upper = max(interval), maximum = FALSE, tol = .Machine$double.eps^0.25)
Arguments
 f
 the function to be optimized. The function is
either minimized or maximized over its first argument
depending on the value of
maximum
.  interval
 a vector containing the endpoints of the interval to be searched for the minimum.
 ...
 additional named or unnamed arguments to be passed
to
f
.  lower
 the lower end point of the interval to be searched.
 upper
 the upper end point of the interval to be searched.
 maximum
 logical. Should we maximize or minimize (the default)?
 tol
 the desired accuracy.
Details
Note that arguments after ...
must be matched exactly.
The method used is a combination of golden section search and
successive parabolic interpolation, and was designed for use with
continuous functions. Convergence is never much slower
than that for a Fibonacci search. If f
has a continuous second
derivative which is positive at the minimum (which is not at lower
or
upper
), then convergence is superlinear, and usually of the
order of about 1.324.
The function f
is never evaluated at two points closer together
than $eps *$$ x_0 + (tol/3)$, where
$eps$ is approximately sqrt(.Machine$double.eps)
and $x_0$ is the final abscissa optimize()$minimum
.
If f
is a unimodal function and the computed values of f
are always unimodal when separated by at least $eps *$
$ x + (tol/3)$, then $x_0$ approximates the abscissa of the
global minimum of f
on the interval lower,upper
with an
error less than $eps *$$ x_0+ tol$.
If f
is not unimodal, then optimize()
may approximate a
local, but perhaps nonglobal, minimum to the same accuracy.
The first evaluation of f
is always at
$x_1 = a + (1\phi)(ba)$ where (a,b) = (lower, upper)
and
$phi = (sqrt(5)  1)/2 = 0.61803..$
is the golden section ratio.
Almost always, the second evaluation is at
$x_2 = a + phi(ba)$.
Note that a local minimum inside $[x_1,x_2]$ will be found as
solution, even when f
is constant in there, see the last
example.
f
will be called as f(x, ...)
for a numeric value
of x.
The argument passed to f
has special semantics and used to be
shared between calls. The function should not copy it.
Value

A list with components
minimum
(or maximum
)
and objective
which give the location of the minimum (or maximum)
and the value of the function at that point.
Source
A C translation of Fortran code http://www.netlib.org/fmm/fmin.f
(author(s) unstated)
based on the Algol 60 procedure localmin
given in the reference.
References
Brent, R. (1973) Algorithms for Minimization without Derivatives. Englewood Cliffs N.J.: PrenticeHall.
See Also
Examples
library(stats)
require(graphics)
f < function (x, a) (x  a)^2
xmin < optimize(f, c(0, 1), tol = 0.0001, a = 1/3)
xmin
## See where the function is evaluated:
optimize(function(x) x^2*(print(x)1), lower = 0, upper = 10)
## "wrong" solution with unlucky interval and piecewise constant f():
f < function(x) ifelse(x > 1, ifelse(x < 4, exp(1/abs(x  1)), 10), 10)
fp < function(x) { print(x); f(x) }
plot(f, 2,5, ylim = 0:1, col = 2)
optimize(fp, c(4, 20)) # doesn't see the minimum
optimize(fp, c(7, 20)) # ok