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Consider $f$ to be a convex multivariate function from $\mathbb{R}^N$ to $\mathbb{R}$, not differentiable (but, by definition, left- and right-differentiable).

Does some kind of "directional" 2nd-order approximation formula exist, for example where $$ f(x+ty) = f(x)+<v,ty>+\frac{1}{2}<Ay,y> $$

for $(x,y)\in\mathbb{R}^N$ and a scalar $t>0$ and $v$ is a subgradient of $f$ at $x$ and $A$ is a matrix? When $f$ is convex not differentiable, I struggle to justify that.

Thanks for your help.

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