# SDP formulation of dual norm

I know that the dual norm of a matrix can be formulated as a semidefinite program (SDP), i.e., $$\|X\|_{2,*}$$ is the solution to the following SDP in $$Y$$:

$$\begin{array}{ll} \text{maximize} & Y^T X\\ \text{subject to} & \begin{bmatrix} I & Y\\ Y^T & I\end{bmatrix} \succeq 0\end{array}$$

If instead of $$X$$, we have $$Z = (I - \phi \phi^T)X$$, where $$P_s = \phi \phi ^T$$ is an orthogonal projection, how can we reformulate our SDP?

\begin{align} \max_{Y,\phi}. \quad Y^TZ \\ s.t. \quad \begin{bmatrix}I & Y\\ Y^T& I \end{bmatrix} \succeq 0 \\ Z = (I - \phi \phi^T)X \\ \phi \succ 0 \end{align}

• Is $\phi$ a variable? If so you can't express this as an SDP, because it's not convex. – Michael Grant Mar 20 at 20:11
• Yes, it is. I was hoping there's a work around perhaps with the schur complement lemma. Thank you for your answer, Michael! – Sanjana Vijayshankar Mar 21 at 15:34