Global optimum

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Polynomial of degree 4, on the right one finds a local optimum, on the left is the 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) = −x² + 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.

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