13.1. Minimization, Maximization

The R function optimize can be used for both minimization and maximization. We start with discussing univariate functions.

13.1.1. Minimization

A univariate function \(f(x)\) is to be minimized over an interval \([a, b]\). The value of the function \(f(x)\) at a particular value of \(x\) is called the objective. The minimization procedure will identify a point \(c \in [a,b]\) such that \(f(c)\) is lower than or equal to all other values of \(f(x) : x \in [a,b]\). The minimization procedure will return:

• The objective value at the point of minimization
• The value of \(x\) at which the minimum is achieved.

Minimizing a function over a given interval:

> f <- function (x) { x^ 2}
> f(2)
[1] 4
> f(-2)
[1] 4
> optimize(f, c(-2, 2))
$minimum
[1] -5.551115e-17

$objective
[1] 3.081488e-33

> f <- function (x) { (x-1)^ 2}
> optimize(f, c(-2, 2))
$minimum
[1] 1

$objective
[1] 0

In the second case, \(f(x)\) is symmetrical around \(x=1\). It is possible to make this parameterized for \(x=a\):

> f <- function (x, a) { (x-a)^ 2}
> optimize(f, c(-10, 10), a=3)
$minimum
[1] 3

$objective
[1] 0

> optimize(f, c(-10, 10), a=4)
$minimum
[1] 4

$objective
[1] 0

> optimize(f, c(-10, 10), a=5)
$minimum
[1] 5

$objective
[1] 0

13.1.2. Maximization

Our function to be maximized:

> f <- function(x, a) {sin(x-a)}

Maximization procedure:

> optimize(f, c(0, pi), a=pi/4, maximum = T)
$maximum
[1] 2.35619

$objective
[1] 1

> pi/4 + pi/2
[1] 2.356194