Quasiconvex function

For the unrelated generalization of convexity used in the calculus of variations, see Quasiconvexity (calculus of variations).

A quasiconvex function that is not convex

A function that is not quasiconvex: the set of points in the domain of the function for which the function values are below the dashed red line is the union of the two red intervals, which is not a convex set.

The probability density function of the normal distribution is quasiconcave but not concave.

The bivariate normal joint density is quasiconcave.

Contour plot of a quasiconvex function (top) where all level sets are convex, and a non-quasiconvex function (bottom) where some level sets are not convex and may even be disconnected.

In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the set of points on which the function value is at most y is a convex set. In other words, the inverse image of any set of the form $(-\infty ,y)$ is a convex set. An equivalent definition is: along any interval in the function domain, the function attains the highest value on one of the endpoints.

Quasiconvexity is a more general property than convexity: all convex functions are also quasiconvex, but not all quasiconvex functions are convex.

For one-dimensional functions (functions on R), to check graphically whether a function is quasiconvex, move a horizontal line from minus infinity upwards, and verify that, whenever the line intersects the region above the function graph, the intersection is an interval.

A quasiconcave function is the negative of a quasiconvex function. In a quasiconcave function, for any real number y, the set of points on which the function value is at least y is convex. Equivalently, along any interval in the function domain, the function attains the lowest value on one of the endpoints. In one dimension, verify that, for any horizontal line that intersects the region below the function graph, the intersection is an interval.

Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex.