Quasiconvex function

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

Equivalently, $f:S \to \mathbb{R}$ is quasiconvex if whenever $x,y \in S$ and $\lambda \in [0,1]$ then $f (λ x + (1 - λ) y)$ is less than or equal to the maximum of $f (x)$ and $f (y)$ .

If $f (λ x + (1 - λ) y)$ is strictly less than the maximum of $f (x)$ and $f (y)$ whenever $x \neq y$ 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.

Many optimization algorithms, that work for convex functions generalize to quasiconvex functions. Two important facts about quasiconvex functions that come into use are:

Any local minimum is a global minimum.
A local maximum can be achieved only on the boundary.

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 quasicovex functions, such as Sion's minimax theorem, which is a far reaching generalization of the result of von Neumann and Morgenstern.