Quasiconvex function

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ${\displaystyle (-\infty ,a)}$ is a convex set. 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.

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

A function ${\displaystyle f:S\to \mathbb {R} }$ defined on a convex subset ${\displaystyle S}$ of a real vector space is quasiconvex if for all ${\displaystyle x,y\in S}$ and ${\displaystyle \lambda \in [0,1]}$ we have

In words, if ${\displaystyle f}$ is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then ${\displaystyle f}$ is quasiconvex. Note that the points ${\displaystyle x}$ and ${\displaystyle y}$, and the point directly between them, can be points on a line or more generally points in n-dimensional space.

An alternative way (see introduction) of defining a quasi-convex function ${\displaystyle f(x)}$ is to require that each sublevel set ${\displaystyle S_{\alpha }(f)=\{x\mid f(x)\leq \alpha \}}$ is a convex set.

If furthermore

for all ${\displaystyle x\neq y}$ and ${\displaystyle \lambda \in (0,1)}$, then ${\displaystyle f}$ is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does.

A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function ${\displaystyle f}$ is quasiconcave if