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Sets and functions are familiar concepts in mathematics. A set is a well-defined collection of distinct objects, considered an object in its own right. A function is a map that takes objects from one set (i.e., input or domain) and returns an object in another set (i.e., output or image). An optimization problem consists of finding the best object in the output set and its corresponding input, as shown schematically in Figure 2.1. The input set Ω is called the set of feasible solutions, and the best object corresponds to a minimum or a maximum, according to an objective function f:Ω⊆Rn→R.

Figure 2.1 Representation of the sets related to a general optimization problem.

Solving an optimization problem implies not only to find the value of the objective function (e.g., f_min or f_max) but also the value x, at the input set Ω (e.g.,x_min, x_max). These values are represented as follows:

(2.1)

(2.2)

Notice that f_min and f_max are numbers whereas x_min and x_max are vectors. The following example shows the difference between min and argmin (the same applies for max and argmax).