nonmaximizable

Nonmaximizable is a term used in optimization to describe an objective—such as a function to be maximized, a cost, or a utility—where a maximum value cannot be attained within the specified domain. In formal terms, a function f defined on a domain D is nonmaximizable if there is no x in D with f(x) ≥ f(y) for all y in D.

There are two common reasons for nonmaximizable objectives. First, the function may be unbounded above on the domain, so no finite maximum exists. Second, the function may be bounded above but fail to attain its maximum due to the domain being non-compact, or because the maximizing point lies on a boundary not included in D, or because the function is discontinuous at that boundary.

Examples help illustrate the idea. Consider f(x) = x on the open interval (0,1). There is no maximizer in (0,1), although the supremum is 1. By contrast, f(x) = 1/x on (0, ∞) is not maximized because the function grows without bound as x approaches 0. These examples show how a problem can have a supremum without a maximum, or have no finite supremum at all.

Implications and approaches: When a problem is nonmaximizable, researchers may consider the supremum as the objective, extend the domain to include a maximizer, or use approximate notions such as epsilon-maximizers. In algorithm design, nonmaximizable problems often require alternative criteria to guarantee convergence or performance.

See also: maximization, optimization, supremum, attainment, compactness, epsilon-optimal.