# 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 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 \(\epsilon\)\( |x_0| + (tol/3)\), 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(`

for a numeric value
of `x`, ...)`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.

##### References

Brent, R. (1973)
*Algorithms for Minimization without Derivatives.*
Englewood Cliffs N.J.: Prentice-Hall.

##### See Also

##### Examples

`library(stats)`

```
# NOT RUN {
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
# }
```

*Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2*