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.

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. 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. 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. 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? His results were later generalized by Connell and Rasmussen, 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.

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