Quasi-convex function
A quasi-convex function is a real-valued function that is defined on a convex subset of a real vector space and that generalizes the property of convex functions that all of their sub-level sets are convex. Similar to the convex functions, the counterpart is defined as the quasi-concave function . If a function is quasi-convex and quasi-concave, it is called a quasi-linear function . Quasi-convex functions are of importance in various applications in economic theory. Optimization methods that are tailored to the class of quasi-convex functions belong to quasi-convex optimization and are generalizations of convex optimization .
definition
Quasi-convex functions can be defined in two ways. Depending on the definition chosen, the other definition is then listed as a property.
About level quantities
A function that is defined on a convex subset S of a real vector space is called
- quasi-convex if any sub-level set
- quasi-concave if any super-level amount
- for anything is convex. Equivalent to this is that is quasi-convex.
- quasi-linear if it is both quasi-convex and quasi-concave.
About inequalities
A function that is defined on a convex subset S of a real vector space is called
- quasi-convex if from and it follows that
- strictly quasi-convex if
- for everyone and applies.
- quasi-concave when from and it follows that
- strictly quasi-concave if
- for everyone and applies.
It is equivalent to the (strict) quasi-concavity of that is (strictly) quasi-convex. The quasi- linearity is defined as above: A function is called quasi-linear if it is quasi-convex and quasi-concave.
Examples
- Every convex function is quasi-convex because the sub-level sets of convex functions are convex.
- Similarly, all concave functions are quasi-concave.
- Every monotonic function is both quasi-convex and quasi-concave, i.e. quasi-linear.
- The rounding function is an example of a quasi-convex function that is neither convex nor continuous.
- Linear functions are quasi-linear.
- is not linear, but quasi-linear.
properties
- Continuous quasi-convex functions on a normalized vector space are always weakly sub-continuous functions .
- Hence, continuous quasi-convex functions take a minimum on weakly sequence compact sets .
- In particular, continuous quasi-convex functions on a convex, closed, restricted and non-empty subset of a reflexive Banach space assume a minimum.
- A continuous function with convex is quasi-convex if and only if at least one of the following three conditions applies:
- is monotone increasing on .
- is monotonously noticeable .
- There is one , so for all monotonically decreasing and for all monotonically increasing.
- The domain and every set of levels of a quasilinear function are convex.
- As with convex functions, a function where is a convex set is quasi-convex if and only if the function is defined by is quasi-convex for all and all directions .
Calculation rules
Pointwise positively weighted maxima
Are quasi-convex functions and positive real numbers for , then is too
a quasi-convex function. This follows from the fact that the sub-level set of the function is exactly the intersection of all sub-level sets of the functions . However, these are by definition convex and thus the level set of as the intersection of convex sets is also convex.
Point-wise supremum
If is a quasi-convex function in for all and is for all , so is also
a quasi-convex function. This can be shown analogously to the case with maxima.
Point-wise infimum
If is quasi-convex both in and in and is where is a convex set , then the function is
quasi-convex.
composition
If the function is quasi-convex and is a monotonically falling function , then is a quasi-convex function.
Quasi-convexity and differentiability
Using the first derivative
The differentiable function is given with convex. Then it is quasi-convex if and only if that applies to all
- .
In the case of a function on the real numbers, this simplifies to
- .
Due to the equivalence, this is also occasionally used to characterize quasi-convexity.
In contrast to convex functions, it follows from quasi-convex functions, or in general not , that there is a minimum . An example of this is the function
- .
It is quasi-convex because it grows monotonically. Its derivative disappears infinitely often, but it has no minimum.
Using the second derivative
If the function is twice differentiable and quasi-convex, then for all and , it follows that . In the case of a function on , this is simplified
Representation by families of convex functions
In the application, one is often interested in modeling level sets of quasi-convex functions by a family of convex functions. This case occurs, for example, with optimization problems with quasi-convex restriction functions. The level sets are convex, but convex functions are easier to use than quasi-convex functions. So we are looking for a family of convex functions for such that
holds for a quasi-convex function . The quasi-convex restriction
can then be determined by the convex restriction
replace. The quasi-convex optimization problem is then a convex optimization problem. is always a monotonically increasing function in , so it is true .
The level quantities are always displayed, for example through the extended function
- .
But it is not clear. Usually one is interested in differentiable functions that describe the level sets.
Applications in economic theory
- In the theory of the household optimum , quasi-concave utility functions appear.
- In the theory of the Nash equilibrium one considers quasi-concave payoff functions.
swell
- M. Avriel, WE Diewert, S. Schaible, I. Zang: Generalized Concavity. Plenum Press, 1988, ISBN 0-306-42656-0 .
literature
- Johannes Jahn: Introduction to the Theory of Nonlinear Optimization . 3. Edition. Springer-Verlag, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-49378-5 .
- Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge / New York / Melbourne 2004, ISBN 0-521-83378-7 ( online ).
Web 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