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 $(-\infty ,a)$ is a convex set. In lay language, if you cut off the top of the function and look down at the remaining part of the function, you always see something that is convex.

All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a weak form of convexity.

Definition and properties

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. Equivalently a function $f$ is quasiconcave if

f(\lambda x+(1-\lambda )y)\geq \min {\big (}f(x),f(y){\big )}.

and strictly quasiconcave if

f(\lambda x+(1-\lambda )y)>\min {\big (}f(x),f(y){\big )}

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.

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.

In economics quasiconcave utility functions are desirable as they imply a strictly convex set of preferences.

Operations preserving quasiconvexity

non-negative weighted maximum of quasiconvex functions (i.e. $f=\max \left\lbrace w_{1}f_{1},\ldots ,w_{n}f_{n}\right\rbrace$ with $w_{i}$ non-negative)
composition with a non-decreasing function (i.e. $g:{\mathcal {R}}^{n}\rightarrow {\mathcal {R}}$ quasiconvex, $h:{\mathcal {R}}\rightarrow {\mathcal {R}}$ non-decreasing, then $f=h\circ g$ is quasiconvex)
minimization (i.e. $f(x,y)$ quasiconvex, $C$ convex set, then $h(x)=\inf _{y\in C}f(x,y)$ is quasiconvex)

Operations not preserving quasiconvexity

sum of quasiconvex functions defined on the same domain (i.e. if $f(x),g(x)$ are quasiconvex, $h(x)=f(x)+g(x)$ could be not quasiconvex)
sum of quasiconvex functions defined on different domains (i.e. if $f(x),g(y)$ are quasiconvex, $h(x,y)=f(x)+g(y)$ could be not quasiconvex)

Examples

Every convex function is quasiconvex.
Any monotonic function is both quasiconvex and quasiconcave. 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.

References

Avriel, M., Diewert, W.E., Schaible, S. and Zang, I., Generalized Concavity, Plenum Press, 1988.

External links

SION, M., "On general minimax theorems", Pacific J. Math. 8 (1958), 171-176.
Mathematical programming glossary
Concave and Quasi-Concave Functions - by Charles Wilson, NYU Department of Economics
Quasiconcave - From Econterms, for About.com
Quasiconcavity and quasiconvexity - by Martin J. Osborne, University of Toronto Department of Economics
Anatomy of Cobb-Douglas Type Utility Functions in 3D - several examples of quasiconcave utility functions