optimize                package: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 'maximum'.

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 '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 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.

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 <URL:
     http://www.netlib.org/fmm/fmin.f> 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:

     'nlm', 'uniroot'.

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

