Quasikonvex

Quasiconvex, or quasikonvex in some texts, refers to a property of real-valued functions defined on a convex subset of a vector space. A function f: C -> R is quasiconvex if for every x and y in C and every t in [0,1], the inequality f(tx + (1−t)y) ≤ max{f(x), f(y)} holds. Equivalently, all sublevel sets {x in C | f(x) ≤ α} are convex for every α in R. This latter formulation is often used because it reduces the property to geometry of level sets rather than the function’s values along a line segment.

Quasiconvexity generalizes convexity. Every convex function is quasiconvex, but not every quasiconvex function is convex. An example of a quasiconvex function that is not convex is a function that equals 0 on a convex interval and 1 outside it; its lower level sets are convex, yet the function itself is not convex. Conversely, many common functions, such as norms, exponentials, and the maximum of finitely many quasiconvex functions, are quasiconvex.

In optimization, quasiconvexity has practical implications: on a convex domain, every local minimum of a quasiconvex function is a global minimum. This makes quasiconvex problems more tractable than general nonconvex problems, although existence of minimizers and algorithmic specifics depend on additional structure like continuity and compactness.

Overall, quasiconvexity is a key concept in optimization and analysis, providing a broader framework than convexity while preserving some favorable minimization properties.