Minorant

A minorant of a function is another function that lies below it. Formally, if f is defined on a domain D, a function g is a minorant of f if g(x) ≤ f(x) for every x in D. Thus g provides a pointwise lower bound for f. Minorants can be real-valued or extended real-valued; the trivial minorant -∞ exists in the extended sense, while discussions of finite-valued minorants focus on real-valued functions.

A related notion is majorant (or upper bound): h is a majorant of f if h(x) ≥ f(x) for all x. A common specialization is an affine (or linear) minorant, a function of the form ax + b that satisfies ax + b ≤ f(x) for all x in the domain. In convex analysis, the greatest affine minorant is particularly important, and more generally the greatest convex minorant of f is its convex envelope.

Examples help illustrate the concept. If f(x) = x^2 on the real line, then m(x) = -|x| is a minorant since -|x| ≤ x^2 for all x. If f is bounded below, the greatest minorant m*(x) = sup{g(x) : g is a minorant of f} exists as a pointwise supremum of all lower bounds, though it may be extended-valued.

Applications of minorants occur in optimization and approximation. They provide lower estimates for objective functions, support lower-bound constructions in branch-and-bound methods, and appear in the formation of envelopes, such as the greatest convex minorant in convex analysis. See also majorant, convex envelope, and subgradient.