Global optimum

In mathematics, a global optimum is a selection from a given domain which yields either the highest value or lowest value (depending on the objective), when a specific function is applied. For example, for the function

:"f"("x") = −"x"² + 2,

defined on the real numbers, the global optimum occurs at "x" = 0, when "f"("x") = 2. For all other values of "x", "f"("x") is smaller.

For purposes of optimization, a function must be defined over the whole domain, and must have a range which is a totally ordered set, in order that the evaluations of distinct domain elements are comparable.

By contrast, a local optimum is a selection for which "neighboring" selections yield values that are not greater. The concept of a local optimum implies that the domain is a metric space or topological space, in order that the notion of "neighborhood" should be meaningful.