nonconvex

Nonconvex is a term used in mathematics to describe objects that fail to satisfy convexity. In its most common usage it refers to either a set or a function. A set S in a real vector space is convex if, for any points x and y in S and any t in [0,1], the point (1−t)x + t y also lies in S. If this property does not hold, S is nonconvex. Likewise a function f is convex if its domain is a convex set and for all x, y in the domain and t in [0,1], f((1−t)x + t y) ≤ (1−t)f(x) + t f(y). If this inequality fails for some pair x, y and t, the function is nonconvex.

Geometrically, nonconvex shapes have indentations, holes, or disjoint parts, such as crescent shapes or annuli. In optimization, nonconvex problems involve objective functions or feasible sets that are not convex, which can lead to multiple local optima and difficulty in finding a global optimum. Many real-world problems in engineering, economics, and machine learning are nonconvex, necessitating heuristics, global optimization methods, or convex relaxations.

Examples include the function f(x) = x^4 − x^2, which is nonconvex near the origin, and feasible regions that exclude holes from a disk. Related concepts include convexity, concavity, quasi-convexity, and convex optimization as a contrasting framework. See also nonconvex optimization.