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

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.

• Strongly convex Aug 1, 2020 at 20:52
• What are the domain and codomain of your function? Usually a Lipschitz operator goes from a space to itself, but strong convexity is a property of a function from a space to $\mathbb{R}$. So are you assuming this operator maps from $\mathbb{R}\to\mathbb{R}$?
– Zim
Aug 2, 2020 at 16:27
• Yes that is right Aug 2, 2020 at 16:29

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.
• Set $x=3$, $y=0$, then $|f(x)-f(y)|=4.5 > 3 = | x-y|$ so $f$ is not 1-Lipschitz. I think the constant is actually $2$, see e.g. math.stackexchange.com/questions/2617575/…
• @Zim you're confusing a Lipschitz function with Lipschitz gradient. The gradient of $\frac{1}{2} x^2 = x$. Then in your example, $3\leq 1*3$. Q.E.D :) Aug 2, 2020 at 16:43
• I agree its gradient is $1$-Lipschitz, but it looks to me like the OP was asking for a strongly convex Lipschitz-continuous function; they did not mention $L$-smoothness at all.
• @Zim you got me there. Since strong convexity and $L$-smoothness are related I might have read it wrong. But we'll need to wait and hear from him. Aug 2, 2020 at 16:57