# What is the dual of the following semidefinite program?

Let us consider the following semidefinite program

\begin{align} \underset{X_{t}, \Omega_t, \Pi}{\text{minimize}} & \quad \frac{1}{T+1} \sum_{t=0}^{T} \text{Tr} (X_t)\\\\ \text{subject to} & \quad \begin{bmatrix} X_{t} & L_t \\ L_t^{\text{T}} & \Omega_{t} \end{bmatrix} \succeq 0, \quad 0 \leq t\leq T, \\\\ & \quad \begin{bmatrix} C_{t+1}^{\text{T}} \Pi C_{t+1} - \Omega_{t+1} + \Xi_{t} & \Xi_{t} A_{t}\\ A_{t}^{\text{T}} \Xi_{t} & \Omega_{t} + A_{t}^{\text{T}} \Xi_{t} A_{t} \end{bmatrix} \succeq 0, \quad 0 \leq t \leq T-1, \\\\ & \quad \begin{bmatrix} I_{p_{i}}/\alpha_{i}^{2} + V_i^{-1} & E_{i}^\text{T}\\ E_{i} & V-V \Pi V \end{bmatrix}\succeq0,\;\; 1 \leq i\leq n. \end{align}

What is its dual? If someone can provide a theoretical process to find it, that will be very helpful. How do I start deriving a dual for a SDP problem? Can you point me to some references please? Thanks.

• You might find this question along with the comments and answer to be helpful. Granted, your situation is more involved. – Omnomnomnom Sep 1 '17 at 1:02
• Grant's comment seems helpful: "Translating to standard form is most certainly not the first step to finding the dual. In fact, you will get the wrong dual if you do so. Yes, it will be "equivalent" in some sense, but it will not be straightforward to map the dual variables you obtain back to the original problem. Instead, you should attach an appropriate Lagrange multiplier to each constraint, and differentiate with respect to each primal variable to find the implicit dual equality constraints." – Omnomnomnom Sep 1 '17 at 1:08