| optimize {stats} | R Documentation |
One Dimensional Optimization
Description
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.
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 |
interval |
a vector containing the end-points of the interval to be searched for the minimum. |
... |
additional named or unnamed arguments to be passed
to |
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 \epsilon |x_0| + (tol/3), where
\epsilon 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 \epsilon
|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 \epsilon |x_0|+ tol.
If f is not unimodal, then optimize() may approximate a
local, but perhaps non-global, minimum to the same accuracy.
The first evaluation of f is always at
x_1 = a + (1-\phi)(b-a) 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(b-a).
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 https://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.: Prentice-Hall.
See Also
Examples
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