Quasi-convex function

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A quasi-convex function that is not convex.
A function that is not quasi-convex: the set of points for which the function values ​​are below the dashed red line is the union of two separate intervals and therefore not convex.

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

The graph of a quasi-concave function.

A function that is defined on a convex subset S of a real vector space is called

for anything is convex .
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

Rounding function or Gaussian bracket function
  • 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:
  1. is monotone increasing on .
  2. is monotonously noticeable .
  3. 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

  1. In the theory of the household optimum , quasi-concave utility functions appear.
  2. 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