In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quasiconvex function is a
real-valued
function defined on an
interval or on a
convex subset of a real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
such that the
inverse image of any set of the form
is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.
All
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
s are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. ''
Univariate''
unimodal
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal p ...
functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple
arguments. For example, the 2-dimensional
Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non- convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Ros ...
is unimodal but not quasiconvex and functions with
star-convex sublevel sets can be unimodal without being quasiconvex.
Definition and properties
A function
defined on a convex subset
of a real vector space is quasiconvex if for all
and