Concavification

In mathematics, concavification is the process of converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical optimization.{{Cite journal|last1=Li|first1=D.|last2=Sun|first2=X. L.|last3=Biswal|first3=M. P.|last4=Gao|first4=F.|date=2001-07-01|title=Convexification, Concavification and Monotonization in Global Optimization|journal=Annals of Operations Research|language=en|volume=105|issue=1–4|pages=213–226|doi=10.1023/A:1013313901854|s2cid=7570136 |issn=0254-5330}}

Concavification of a quasiconcave function by monotone transformation

An important special case of concavification is where the original function is a quasiconcave function. It is known that:

Every concave function is quasiconcave, but the opposite is not true.
Every monotone transformation of a quasiconcave function is also quasiconcave. For example, if $f : \mathbb{R}^n \to \mathbb{R}$ is quasiconcave and $g : \mathbb{R} \to \mathbb{R}$ is a monotonically-increasing function, then $x \mapsto g(f(x))$ is also quasiconcave.

Therefore, a natural question is: given a quasiconcave function $f : \mathbb{R}^n \to \mathbb{R}$ , does there exist a monotonically increasing $g : \mathbb{R} \to \mathbb{R}$ such that $x \mapsto g(f(x))$ is concave?

= Example and Counter Example =

As an example, consider the function $x \mapsto f(x) = x^2$ on the domain $x\geq 0$ . This function is quasiconcave, but it is not concave (in fact, it is strictly convex). It can be concavified, for example, using the monotone transformation $t \mapsto g(t) = t^{1/4}$ , since $x \mapsto g(f(x))=\sqrt{x}$ is concave.

Not every concave function can be concavified in this way. A counter example was shown by Fenchel.{{Cite book|title=Convex cones, sets and functions|last=Fenchel|publisher=Princeton University|year=1953}} His example is: $(x,y) \mapsto f(x,y) := y + \sqrt{x+y^2}$ . Fenchel proved that this function is quasiconcave, but there is no monotone transformation $g : \mathbb{R}\to\mathbb{R}$ such that $(x, y) \mapsto g(f(x,y))$ is concave.{{Rp|7–9}}

Based on these examples, we define a function to be concavifiable if there exists a monotone transformation that makes it concave. The question now becomes: what quasiconcave functions are concavifiable?

= Concavifiability =

Yakar Kannai treats the question in depth in the context of utility functions, giving sufficient conditions under which continuous convex preferences can be represented by concave utility functions.{{Cite journal|date=1977-03-01|title=Concavifiability and constructions of concave utility functions|journal=Journal of Mathematical Economics|language=en|volume=4|issue=1|pages=1–56|doi=10.1016/0304-4068(77)90015-5|issn=0304-4068|last1=Kannai|first1=Yakar}}

His results were later generalized by Connell and Rasmussen,{{Cite journal|last1=Connell|first1=Christopher|last2=Rasmusen|first2=Eric Bennett|date=December 2017|title=Concavifying the QuasiConcave|journal=Journal of Convex Analysis|language=en|volume=24|issue=4|pages=1239–1262}} who give necessary and sufficient conditions for concavifiability. They show that the function $(x,y) \mapsto f(x,y) = e^{e^x}\cdot y$ violates their conditions and thus is not concavifiable. They prove that this function is strictly quasiconcave and its gradient is non-vanishing, but it is not concavifiable.

References

Category:Convex analysis