nearmaximal

Nearmaximal is an adjective used across disciplines to describe an object that is close to being maximal but does not actually attain a maximum. Because there is no universal standard definition, the precise meaning of nearmaximal is determined by the context and the structure involved (for example, a set, a function, or a partially ordered set). Commonly, nearmaximal connotes quantitative closeness to a maximum rather than qualitative closeness to a property.

In optimization and real analysis, a typical usage is in terms of a real-valued function f defined on a domain D. Let M be the supremum of f over D. An element x in D is called ε-near-maximal if f(x) ≥ M − ε for a given tolerance ε > 0. This characterization is often used to describe approximately optimal solutions, sometimes called near-optimal. The term nearmaximal can thus be translated into standard language depending on the tolerance used and the available information about M.

In order theory and topology, nearmaximality may be used more loosely. It can describe an element that is maximal within a restricted substructure, or a point that lies in the closure of the set of maximal elements with respect to a given topology. These uses emphasize proximity to the top of the order or to the limit of maximal elements, rather than a strict maximum.

Notes: Because nearmaximal is not a rigorously standardized term, readers should consult the definitions provided in the relevant work. Related terms include maximal element, maximum, near-optimal, and approximation.