Critical Value Functions have Finite Modulus of Concavity

Abstract

We consider a smooth finite dimensional parametric optimization problem ${\cal P}(y)$ with objective function $f(x,y)$. Here, x and y denote the state variable and the parameter, respectively. In the case that $\overline{x}$ is a strongly stable Karush--Kuhn--Tucker point for ${\cal P}(\overline{y})$, a neighborhood of $\overline{x}$ contains a unique Karush--Kuhn--Tucker point $x(y)$ for ${\cal P}(y)$, provided that y is sufficiently close to $\overline{y}$. This gives rise to the critical value function $y\mapsto\varphi(y):=f(x(y),y)$. Under the additional assumption that the Mangasarian--Fromovitz constraint qualification is satisfied at $\overline{x}$, we show that $\varphi$ has finite modulus of concavity. That means $\varphi$ becomes convex in a neighborhood of $\overline{y}$ by adding to it the function $y\mapsto (\alpha/2)\cdot\|y-\overline{y}\|^2$ for some $\alpha>0$. Moreover, we present an explicit upper bound for the $\alpha$ to be used. The latter bound turns out to be sharp for problem data in general position.