Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 33

The function f(x) = x² is not monotone; for example −3 ≤ 1 but f(−3)≰f(1). Nevertheless, the function is monotone increasing in R+₊. In this set, 4 ≤ 8 implies that f(x) ≤ f(y) since both 4 and 8 belong to R+₊.

An ordered set Ω ∈ ⁿ admits the following definitions:

 Supreme: the supreme of a set, denoted by sup(Ω), is the minimum value greater than all the elements of Ω.

 Infimum: the infimum of a set, denoted by inf(Ω), is the maximum value lower than all the elements of Ω.

The supreme and the infimum are closely related to the maximum and the minimum of a set. They are equal in most practical applications. The main difference is that the infimum and the supreme can be outside the set. For example, the supreme of the set Ω = {x : 3 ≤ x ≤ 5} is 5 whereas its maximum does not exists. It may seem like a simple difference, but several theoretical analyzes require this differentiation.

Some properties of the supreme and the infimum are presented below:

(2.7)

(2.8)

(2.9)

(2.10)

Moreover, the last case implies that:

(2.11)

That is to say, the value of x that minimizes the function f(x) + α is the same value that minimizes f(x); for this reason, it is typical to neglect the constant α in practical problems.