I would like to solve $$\min_{w}||Bw||*{\infty}+\frac{1}{2}||w-\eta||*{2}^{2}$$ where w=(w_{1},w_{2})\in(\mathbb{R}^{n\times m})^{2} and $$||Bw||*{\infty}=\max*{i,j}(B_{i,j}\circ|w|)$$ where |w|=\sqrt{w_{1}^{2}+w_{2}^{2}}. It is very easy and efficient to solve this when B is a scalar which reduces to a projection onto \ell_{1} ball (proximity identity). Is it possible to find an exact solution when B is a matrix? I can do it using CVX but since the above minimization is an inner subproblem it is not efficient.

Thanks in advance!!!