- Quasiconvex function
In
mathematics , a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a realvector space such that theinverse image of any set of the form infty,a) is aconvex set .Definition and properties
Equivalently, a function f:S o 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)leqmaxig(f(x),f(y)ig).
If instead
: f(lambda x + (1 - lambda)y)
for any x eq 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.
A (strictly) quasiconvex function has (strictly) convex
lower contour set s, while a (strictly) quasiconcave function has (strictly) convexupper contour set s.Optimization methods that work for quasiconvex functions come under the heading of
quasiconvex programming . This comes under the broad heading ofmathematical programming and generalizes bothlinear programming andconvex programming .There are also
minimax theorem s on quasiconvex functions, such asSion's minimax theorem , which is a far-reaching generalization of the result ofvon Neumann andOskar Morgenstern .Examples
* Every convex function is quasiconvex.
* Anymonotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex.
*Thefloor function xmapsto lfloor x floor is an example of a quasiconvex function that is neither convex nor continuous.ee also
*
Convex function
*Pseudoconvex function References
* Avriel, M., Diewert, W.E., Schaible, S. and Zang, I., "Generalized Concavity", Plenum Press, 1988.
External links
* [http://projecteuclid.org/euclid.pjm/1103040253 SION, M., "On general minimax theorems", Pacific J. Math. 8 (1958), 171-176.]
* [http://glossary.computing.society.informs.org/second.php Mathematical programming glossary]
* Charles Wilson, NYU Department of Economics, "Concave and Quasi-Concave Functions": http://www.wilsonc.econ.nyu.edu/UMath/Handouts/ums06h23convexsetsandfunctions.pdf
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