Quasiconvex function

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

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

A quasiconcave function is a function whose negative is 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 Meumann and Morgenstern.

See also

• Convex function
• Strongly quasiconvex function
• Strictly quasiconvex function
• Pseudoconvex function

External links

Mathematical programming glossary