# How to solve the following optimization problem with projection?

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.

• Could you explain what each variable is? It doesn't seem to be a function which returns a scalar. – Royi Jun 7 '16 at 10:04
• Question edited. – user85361 Jun 7 '16 at 14:21
• 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... – Royi Jun 7 '16 at 16:44