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How to solve the following optimization problem with projection? \begin{alignat}{1} &\min_{u_+,u_-,s,l\geq 0} \frac{1}{\lambda} \langle A ,(a +u_+-u_-)(a +u_+-u_-)^\mathsf{T} \rangle+\mathbf{1}^\mathsf{T} s + c l \cr &\text{s.t } u_++u_-=s+l \mathbf{1} \end{alignat} Where $A$ is a constant positive semidefinite matrix. I need some close form solution which iteratively converges to solution.

Can we use Proximal methods for this problem? How?

$A$ is an $n\prod n$ Matrix. $u_+$, $u_-$,$s$ are $R^n$ vector and $l$ is scalar.

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  • $\begingroup$ Could you explain what each variable is? It doesn't seem to be a function which returns a scalar. $\endgroup$ – Royi Jun 7 '16 at 10:04
  • $\begingroup$ Question edited. $\endgroup$ – user85361 Jun 7 '16 at 14:21
  • $\begingroup$ Still, I don't understand what's $ {u}_{+} $ for instance. What's $ {\mathbf{1}}^{T} $ and $ cl $ (Is is c multiplying l or what?) etc... $\endgroup$ – Royi Jun 7 '16 at 16:44

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