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### Local optima of non-negative-least-squares?

Broadly speaking, with convex problems, necessary conditions are sufficient. A nice characterisation of an optimal solution is: If the cost $f$ is convex & differentiable and the constraint set $C$...
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If your vector $x$ has $n$ components, then simply set all $x_i:=\frac{1}{n}$. Then $\bar{x}=\frac{1}{n}$, each $x_i$ is equal to this mean, your sum of squares is zero (so it is minimal, because ...