quasiconcave

Quasiconcave is a property of real-valued functions defined on a convex domain. A function f: D → R is quasiconcave if for all x, y in D and for every t in [0, 1], f(t x + (1−t) y) ≥ min{f(x), f(y)}. Equivalently, for every α ∈ R, the upper level set {x ∈ D | f(x) ≥ α} is convex. If the inequality is strict for 0 < t < 1 and x ≠ y with f(x) ≠ f(y), f is strictly quasiconcave.

Quasiconcavity sits between concavity and the broader class of functions. Every concave function is quasiconcave, but not every quasiconcave function is concave. The related notion of quasiconvex uses lower level sets to be convex. A common operation is to apply an increasing transformation: if φ is increasing and f is quasiconcave, then φ∘f is quasiconcave.

Examples help illustrate the concept. Linear functions are both quasiconcave and quasiconvex. The function f(x) = min{x1, x2} on R^2 is quasiconcave since its upper level sets {f ≥ α} equal [α, ∞) × [α, ∞), a convex set. The negative of a convex function is quasiconcave; for instance f(x) = −||x|| is quasiconcave on R^n because its upper level sets are balls, which are convex.

In optimization, quasiconcavity is a useful property: on a convex domain, any local maximum of a quasiconcave function is a global maximum. This makes certain nonconvex-looking problems tractable under appropriate conditions and is widely used in economics and optimization theory.