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While this is "locally strongly convex" away from $x=0$, its "local modulus of strong convexity" decreases to zero for $x\to 0$. …

I doubt that there is a simple bound without some other restriction. Consider $f(x) = C|x|$, i.e. $\partial f(0) = [-C,C]$. Then there is a convex $g$ with $\|f-g\|_\infty\leq\epsilon$ but $g\equiv 0$ …

For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$: \begin …

My prime example of such an operator comes from saddle point problems of the form $$ \min_x\max_y F(x) + \langle Kx,y\rangle - G(y) $$ with $F,G$ being two proper, convex, lower-semicontinuous functio …

I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) bu …

Note that the conclusion is that the gradient of $g_r$ converges to zero, not $g_r$ itself! The first inequality and the fact that $r>0$ imply $$\ g_r(M(y^k)) - g_r(y^k) > r|\nabla g_r(y^k)|^2 \geq 0 …

I second Czenek's recommendation. Moreover, it could be helpful to know this cute little characterization of invexity (due to Craven and Grower, see also here): Theorem: A differentiable function $f$ …