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.