Envelope theorem for second derivative

Question

I am maximizing a function $f(x,z)$ on $x$ ($z$ is treated a parameter in the maximization). The function $f$ is strictly concave on both variables.

I know how to use the envelope theorem for the first derivative. But I am interested in knowing if the function $f$ evaluated at the maximum, i.e., $f(x^*(z),z)$, where $x^*(z)$ is the solution of the maximization, is also a concave function of $z$.

Is there any theorem for this?

Answer 1

Yes, the resulting function is concave in $z$. This operation is sometimes called partial maximization and is known to preserve concavity (strict concavity is not required).

More precisely, if $f(x,z)$ is a concave function, then the partially maximized function $g(z)$

$g(z) = \sup_{x} f(x,z)$

is also concave.

This can be found for example in the Convex Optimization book of Boyd and Vandenberghe (see Chapter 3.2.5, written for the slightly different but equivalent operation of partial minimization of a convex function).