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I have a question. The answer may be too obvious but I cannot be sure about the right answer. Let say that we have a convex but non-smooth function which is defined as $f : \mathbb R^2 → \mathbb R$. For a point in $\mathbb R^2$, if we know that the function increases for the directions $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$, can we sure about that this point is local optimum (because of the convexity it will be also global optimum)?

Thank you in advance.

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No. Consider for instance $f(x,y) = x^2 + y^2 + 10|y-x|$ at, say, $(1,1)$.

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