My textbook describes smooth optimization problems as "problems where the objective function and constrains set are represented using continuously differentiable functions". Then for example, if I have a problem with the constraint set x ≥ 0, y ≥ x, would it be smooth (if objective function is continuously differentiable)?
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Yes, your example is considered a smooth problem. Both constraints are continuously differentiable functions.
I assume you have some confusion since you think of the point (0,0) as being non-differentiable. However, a multi-variable function is considered continuously differentiable if the gradient is a continuous map with respect to each variable separately. This is not to be mistaken with Differentiability, which is a stronger condition, but this is not required so that an optimization problem is smooth.
iarbel84
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thank you for the answer! I am more confused with finding gradient of inequalities? – judie Sep 06 '20 at 21:55
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An inequality defines a sublevel set, which is simply a set. But without going too much into details, just make sure your inequality is of the form $g(x)\leq 0$. Then if $\nabla g(x)$ is a continuous map (as I defined above), we can say the constraint is smooth. Hope that helps. – iarbel84 Sep 07 '20 at 09:01