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I have recently been reading about optimization and have come across two (seemingly different) notions of "L-smoothness". Let me record them now; let $H$ be a separable Hilbert space and $f:H \rightarrow (-\infty,\infty)$ be Fréchet differentiable and fix $L>0$.

L-Smoothness - Definition 1 $f$ is called L-smooth if: for every $x_1,x_2\in H$ $$ \frac{L}{2}\|\cdot\|_H^2 - f \mbox{is convex}. $$

here is the second definition

L-Smoothness - Definition 2 $f$ is called L-smooth if its Fréchet derivative $\nabla f$ is $L$-Lipschtiz.

Why/hpw are these two equivalent?

Out of curiosity: How do things generalize to separable reflexive Banach spaces?

AB_IM
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  • I believe the first is usually referred to as strong convexity (albeit what I have seen is that $x \mapsto f(x)-{\mu \over 2} |x|^2$ is convex). – copper.hat Mar 21 '22 at 22:28
  • @copper.hat Yes exactly, somehow strong convexity is the lower-bound version of L-smoothness – AB_IM Mar 21 '22 at 22:39
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    Both are equivalent. See the book of Bauschke & Combettes, Thm 18.15 – daw Mar 22 '22 at 16:16

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