Example of a strongly convex function where the Lipschitz constant $L$ is equal to the strong convexity parameter $u$

Question

I am trying to come up with like three strongly convex function $f\colon\mathbb{R}\to\mathbb{R}$ where the Lipschitz constant $L$ is equal to the strong convexity parameter $u$, i.e. for every $x,y\in\mathbb{R}$ \begin{align} |\nabla f(x)-\nabla f(y)|&\leq L|x-y|, \tag{1}\\ f(\cdot)-\dfrac{u}{2}|\cdot|^2&\;\;\text{is convex, and} \tag{2}\\ L=u \tag{3} \end{align}

I am having a hard time coming up with such functions but I am suspecting that a type of least square might possess the property. Can someone help me come up with such functions where $L=u$? Thanks.

Answer 1

It holds for $f(\mathbf{x})=\frac{1}{2}\|\mathbf{x}\|_2^2$, which has a strong convexity and L-smoothness parameter $1$ w.r.t the $l_2$ norm.

We derive this from the Conjugate Correspondence Theorem which states that a $\mu$-strongly convex function has a conjugate $f^*$ which is $\frac{1}{\mu}$-smooth. Since we have the "rare" occasion where $\frac{1}{2}\|\mathbf{x}\|_2^2$ is it's own conjugate, with the parameter $1=1^{-1}$, the two coincide.