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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.