Semiinfinite

Semiinfinite is an adjective used in mathematics, physics, and engineering to describe an object or domain that extends indefinitely in one direction while having a finite extent in another. The term highlights a half-infinite nature, such as a starting point with infinite continuation.

In geometry and real analysis, a semiinfinite line, or ray, consists of points of the form p + t v with t ≥ 0 for a fixed point p and direction v; it begins at p and extends to infinity along v. A semiinfinite interval on the real line is of the form [a, ∞) or (-∞, a], including the endpoint a. These objects contrast with finite lines or intervals and with fully infinite constructs.

In optimization, semi-infinite programming refers to problems with finitely many decision variables but infinitely many constraints, typically written as minimize f(x) subject to g(x, t) ≤ 0 for all t in an infinite index set T. Such problems demand methods capable of handling an uncountable or otherwise large constraint set.

In physics and materials science, semiinfinite concepts appear in the semiinfinite potential well in quantum mechanics, where a particle encounters an impenetrable barrier on one side, and in the notion of a semiinfinite solid (half-space) used to model boundary and surface effects in elasticity, heat conduction, and diffusion.