| optimize {stats} | R Documentation |
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.
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, ...)
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 end-points of the interval to be searched for the minimum. |
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. |
... |
additional arguments to f. |
The method used is a combination of golden section search and
successive parabolic interpolation. 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 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.
It uses a C translation of Fortran code (from Netlib) based on the
Algol 60 procedure localmin given in the reference.
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.
Brent, R. (1973) Algorithms for Minimization without Derivatives. Englewood Cliffs N.J.: Prentice-Hall.
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), l=0, u=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