how to prove this conclusion:

Suppose that $f$ is a smooth convex function with Lipschitz continuous gradient on $X$, then there exists a self-adjoint and positive semi-definite linear operators $\sum_{f}$ such that for any $x, x^{'}\in X$, $$f(x)\geq f(x') + \langle x -x', \nabla f(x')\rangle + 0.5 *\|x-x'\|_{\sum_{f}}^2.$$

Thanks for your attention and response.


What about the function $f(x) = 0$ !!!

  • $\begingroup$ In your case, $\sum_{f} = 0$. $\endgroup$ – Kai Tu Oct 6 '18 at 10:58
  • $\begingroup$ @KaiTu you said "semi" though ! Did you mean positive definite ? $\endgroup$ – Red shoes Oct 6 '18 at 19:44

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