Quasiconvex function

A quasiconvex function is a real-valued function defined on a convex portion of a real vector space, if all sets of the form

So all preimages of the sets are convex. The concept of quasiconvex functions is therefore a generalization of the concept of convex function; it is of importance in various applications in economic theory. Optimization methods that are tailored to the class of quasiconvex functions, belong to the quasiconvex optimization and are generalizations of the convex optimization.

Definition and properties

Equivalent may be defined: a function that is defined on a convex portion of a real vector space S is, quasiconvex, and if it follows that

If, instead,

For all and is then called strictly quasiconvex.

Quasikonkave function

A quasikonkave function is a function f whose negative -f is quasiconvex, and a strictly quasikonkave function is a function f for which f is strictly quasiconvex. Quasikonkave functions can also be defined by the inequality

By for all and define, and strict Quasikonkavität

For all and.

In a ( strict) quasikonkaven function, the quantities

( strictly ) convex.

Examples

• Every convex function is quasiconvex.
• Any monotonic function is both quasiconvex and quasikonkav.
• Any function that monotonically increases to a certain point and decreases monotonically from the point is quasikonkav.
• The rounding function is an example of a quasiconvex function is neither convex nor continuous.

Applications in economic theory

Swell

• Avriel, M., Diewert, WE, Schaible, S. and Zang, I., Generalized Concavity, Plenum Press, 1988.