How to prove this property of Lipschitz?

Question

Given a convex and twice differentiable function $f$, and $\nabla f$ is Lipschitz with constant $L$, how to prove that $(\nabla f(x) - \nabla f(y))^T(x-y) \leq L \|x-y\|_2^2$ for all $x,y$

I just stuck at $|f(x)-f(y)||x-y| \leq L\|x-y\|^2_2$

Answer 1

From Cauchy-Schwarz inequality, we have

\begin{align} (\nabla f(x) - \nabla f(y))^T(x-y) & \leq \left\| \nabla f(x)-\nabla f(y)\right\|\left\|x-y \right\| \\ &\leq L\left\| x-y\right\|\left\| x-y\right\| \\ &=L \left\| x-y\right\|^2 \end{align}

where the second inequality is due to $\nabla f$ is Lipschitz.