Nominimum

Nominimum is an informal term occasionally used in mathematics and computer science to describe a situation in which a function or sequence has no minimum value that is actually attained within its domain. The word is not part of standard technical vocabulary; more rigorous writers say that the function has no minimum or that the minimum is not attained.

Definition and distinction

In a function f: D → R, f is said to have a nominimum on D if there is no x in D such that f(x) is the smallest value among all f(y) with y in D. Equivalently, there is no argmin of f on D. A nominimum does not preclude the existence of an infimum: the greatest lower bound of f(D) may exist but be unattained. It also covers cases where the function is unbounded below, in which case no minimum exists for sure.

Examples

A classic example is f(x) = e^x on the real line. For every x, you can find y with f(y) < f(x), so no minimum is attained; the infimum is 0, not attained. Another example is f(x) = x on the open interval (0, ∞); the infimum is 0, which is not attained, so f has a nominimum on that domain.

Relation to standard terms

When precision is required, mathematicians would simply say that a function has no minimum or that the minimum is not attained. Nominimum can serve as an informal shorthand in discussions of optimization, analysis, or algorithmic convergence, but it is not a formal concept in most textbooks. See also minimum, maximum, attainment, and infimum.