# Optimization Problem Involving Frobenious Norm, Linear Equality Constraints and Semi Orthogonal Matrix

I have a question, Suppose $$A, B, D \in \mathbb{R}^{n \times p}$$, I want to solve the following optimization problem:

$$\arg\min_D\frac{1}{2}\|D + A\|_F^2 + \lambda\|BB^T + DB^T + BD^T\|_1 \mbox{ s.t. } D^TB + B^TD = 0, B^TB = I_p$$.

We can view that $$D$$ is on the tangent space of Stiefel Manifold.

Do you guys have any idea on how to solve that?

Thanks a lot!

• I don't think this is a Convex Optimization problem as I'm not sure about the convexity of $\mathcal{M} = \left\{ M \mid {M}^{T} M = I \right\}$. – Royi Jan 24 at 11:53
• – Royi Jan 24 at 12:37
• If we minimize with respect to $D$, how come you put the constraint ${B}^{T} B = I$? It should be given. – David Jul 27 at 16:48