Miinimumi

Miinimumi is a term used in theoretical discussions of optimization to denote the collection of points at which a real-valued objective function attains its smallest value on a specified domain. It is analogous to the conventional minimum but is often employed to emphasize the potential multiplicity of minimizers or the context of nonstandard domains.

Formally, let F be a nonempty domain and f: F → R. A point x* ∈ F is a miinimumi of f on F if f(x*) ≤ f(x) for all x ∈ F. The miinimumi set, Mi(f, F), consists of all x ∈ F that satisfy this condition. If Mi(f, F) contains a single element, the minimum is unique; if it contains multiple elements, there are several minimizers.

Existence of a miinimumi depends on the properties of the domain and function. If F is compact and f is continuous on F, Mi(f, F) is nonempty. In finite domains, a miinimumi always exists. In general, Mi(f, F) may be empty when the function has no point attaining its smallest value, even if the infimum is finite, or when it is unbounded below.

Examples illustrate the concept. For f(n) = (n−3)^2 on F = Z, Mi(f, F) = {3}. For f(x) = x^2 on F = [0,1], Mi(f, F) = {0}. For f(x) = −x^2 on F = R, Mi(f, F) is empty, even though the function decreases without bound.

Miinimumi appears chiefly in expository contexts and in discussions of optimization where the entire set of minimizers is of interest, such as in multi-solution problems. See also minimum, infimum, minimization, optimization.