# Smoothness of the function $\lambda \mapsto \inf_{x \in \mathcal{C}}\|\lambda x - y\|^2$

Let $$y\in \mathbb{R}^d$$ and suppose that $$\mathcal{C}\subset \mathbb{R}^d$$ is a compact convex set. I want to know if the function $$\psi\colon\lambda \mapsto \inf_{x \in \mathcal{C}}\|\lambda x - y\|^2$$ is smooth/Lipschitz-continuous on the interval $$[0,1]$$ in the sense that $$\psi$$ is differentiable and there exists some $$L>0$$ (that may depend on $$\mathcal{C}$$ and $$y$$) such that \begin{align} |\psi'(\lambda)- \psi'(\gamma)|\leq L |\lambda - \gamma|, \quad \text{for all } \lambda, \gamma \in [0,1]. \end{align} What I know: The function $$\psi$$ can be written as the (squared) perspective transform of the function $$\varphi\colon y \mapsto \inf_{x\in \mathcal{C}}\|x - y\|$$. That is, $$\psi(\lambda)= (\lambda\cdot \varphi(y/\lambda))^2$$. This implies that I) $$\psi$$ is convex since the perspective transform of a convex function is convex and the composition of a positive convex function with $$w \mapsto w^2$$ is convex; and II) that $$\psi$$ is differentiable on $$(0,1]$$, since $$\varphi^2$$ is differentiable. The latter follows by the fact that $$\varphi^2$$ is the Moreau-Yosida envelope of the indicator function of the set $$\mathcal{C}$$, which Moreau showed is $$C^1$$.

Ultimate goal: My ultimate goal is to be able to numerically find the minimum of the function $$\Theta\colon \lambda \mapsto\lambda^2 + \psi(\lambda)$$ in the interval $$[0,1]$$ very efficiently (for different $$y$$'s). The function $$\Theta$$ is 1-strongly convex (since $$\psi$$ is convex). But it seems that I am missing smoothness/Lipschitz-continuity to be able to show very fast (e.g. linear) convergence to the minimum using known optimization algorithms.

• Did you try to compute $\psi'$ via the chain rule? Maybe one can check the Lipschitzness of the derivative directly?
– gerw
May 11, 2022 at 9:57