Quasiconvex function

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A quasiconvex function which is not convex.

A function which 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.

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 $(-\infty,a)$ is a convex set.

1 Definition and properties
2 Examples
3 See also
4 References
5 External links

[edit] Definition and properties

Equivalently, a function $f:S \to \mathbb{R}$ defined on a convex subset S of a real vector space is quasiconvex if whenever $x,y \in S$ and $\lambda \in [0,1]$ then

$f(\lambda x + (1 - \lambda)y)\leq\max\big(f(x),f(y)\big).$

If instead

$f(\lambda x + (1 - \lambda)y)<\max\big(f(x),f(y)\big)$

for any $x \neq y$ and $\lambda \in (0,1)$ , then $f$ is strictly quasiconvex.

A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex.

A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets.

Optimization methods that work for quasiconvex functions come under the heading of quasiconvex programming. This comes under the broad heading of mathematical programming and generalizes both linear programming and convex programming.

There are also minimax theorems on quasiconvex functions, such as Sion's minimax theorem, which is a far-reaching generalization of the result of von Neumann and Oskar Morgenstern.

[edit] Examples

Every convex function is quasiconvex.
Any monotonic function is quasiconvex. More generally, a function which decreases up to a point and increases from that point on is quasiconvex.
The floor function $x\mapsto \lfloor x\rfloor$ is an example of a quasiconvex function that is neither convex nor continuous.