Quasiconvex function

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.

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 graph of a quasiconcave function.

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.

See also

• Convex function
• Pseudoconvexity
• Pseudoconvex function
• Concave function

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