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A convex set includes all points lying between any two points from the set. A convex function on such a set is a function lying below any straight line connecting two points from its graph. Convex optimization is concerned with searching for the minimum of such a function.
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Partial derivative of a Group Lasso
The objective function is assumed to be differentiable and convex, and the $\|X_{G_l}\|$ can be $=0$ and $\neq 0$. …