Let us consider four simple optimization problems and their respective solution, namely1:
As aforementioned, the operator min returns a number while the operator argmin can return a vector. Two or more points could produce the same minimum. In that case, the argmin is not unique.
But, what exactly does it mean the best solution? And, what characteristics should have both the sets and the functions involved in the problem? Some mathematical sophistication is required to answer these questions. Finding the best solution in a set implies comparing one element with the rest of the set elements. A comparison is a relation of the form x ≤ y or x ≥ y. However, not all sets allow these types of comparisons; those that enable it are called ordered sets. For instance, the real numbers and the integer numbers are all ordered set. However, complex numbers are non-ordered because such a comparison is not possible (what number is higher: zA = 1 + j or iB = 1 − j?).
The objective function establishes a criterion of comparison. Therefore, its output must be an ordered set. Nevertheless, the input set may be ordered or non-ordered; it depends on the problem’s representation. For example, the optimal power flow in power distribution networks targets an ordered set since the active power losses belongs to the real numbers; however, the input may be represented as a vector (v,θ)∈R2×n or as a set of phasors vejθ∈Cn). The former is an ordered set, whereas the latter is non-ordered. Other possible representations can be the set of positive definite matrices or positive polynomials, as presented in Chapter 10. In conclusion, the objective function must point to an ordered set, but the input set (i.e., the set of feasible solutions) can be any arbitrary set.
We usually compare values in the output set since our objective is to minimize or maximize the objective function. It is also possible to compare values in Ω when it is an ordered set. However, a comparison between elements of the input set may be different in the output set. A function f is monotone (or monotonic) increasing, if x ≤ y implies that f(x) ≤ f(y), that is to say, the function preserves the inequality. Similarly, a function is monotone decreasing if x ≤ y implies that f(x) ≥ f(y), that is to say, the function reverses the identity.
Example 2.2
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 ∈
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:
Moreover, the last case implies that:
