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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.

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

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