Quasiconvex function

A function ${\displaystyle f:S\to \mathbb {R} }$ defined on a convex subset ${\displaystyle S}$ of a real vector space is quasiconvex if for all ${\displaystyle x,y\in S}$ and ${\displaystyle \lambda \in [0,1]}$ we have

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

In words, if ${\displaystyle f}$ is such that it is always true that a point directly between two other points does not give a higher value of the function than both of the other points do, then ${\displaystyle f}$ is quasiconvex. Note that the points ${\displaystyle x}$ and ${\displaystyle y}$, and the point directly between them, can be points on a line or more generally points in n-dimensional space.

A quasilinear function is both quasiconvex and quasiconcave.

The graph of a function that is both concave and quasiconvex on the nonnegative real numbers.

An alternative way (see introduction) of defining a quasi-convex function ${\displaystyle f(x)}$ is to require that each sublevel set ${\displaystyle S_{\alpha }(f)=\{x\mid f(x)\leq \alpha \}}$ is a convex set.

If furthermore

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

for all ${\displaystyle x\neq y}$ and ${\displaystyle \lambda \in (0,1)}$, then ${\displaystyle f}$ is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does.

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 ${\displaystyle f}$ is quasiconcave if

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

and strictly quasiconcave if

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

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.

A particular case of quasi-concavity, if ${\displaystyle S\subset \mathbb {R} }$, is unimodality, in which there is a locally maximal value.

Quasiconvex functions have applications in mathematical analysis, in mathematical optimization, and in game theory and economics.

• Every convex function is quasiconvex.
• A concave function can be quasiconvex. For example, ${\displaystyle x\mapsto \log(x)}$ is both concave and 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 (compare unimodality).
• The floor function ${\displaystyle x\mapsto \lfloor x\rfloor }$ is an example of a quasiconvex function that is neither convex nor continuous.

• Convex function
• Concave function
• Logarithmically concave function
• Pseudoconvexity in the sense of several complex variables (not generalized convexity)
• Pseudoconvex function
• Invex function
• Concavification

• Avriel, M., Diewert, W.E., Schaible, S. and Zang, I., Generalized Concavity, Plenum Press, 1988.
• Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 815–816. doi:10.1057/9780230226203.1375. ISBN 978-0-333-78676-5.
• Singer, Ivan Abstract convex analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997. xxii+491 pp. ISBN 0-471-16015-6

• 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
• Quasiconcavity and quasiconvexity - by Martin J. Osborne, University of Toronto Department of Economics