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Concept
Quasiconvex function
Summary
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 of any set of the form 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.
All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. 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 defined on a convex subset of a real vector space is quasiconvex if for all and we have
In words, if 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 is quasiconvex. Note that the points and , 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 is to require that each sublevel set
is a convex set.
If furthermore
for all and , then 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 is quasiconcave if
and strictly quasiconcave if
A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets.
A function that is both quasiconvex and quasiconcave is quasilinear.
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