Dual of a SDP primal with an extra constraint

Question

Consider the following (standard) primal SDP:

$$ \begin{aligned} \min_{} & \quad \langle{C,X}\rangle \\ {\rm s.t.} & \quad \langle A_i,{X}\rangle = b_i, \\ & \quad X \succeq 0,\\ &\quad Y \succeq 0\ \end{aligned} $$ Except for last PSD constraint, the previous is the standard formulation of a primal SDP problem with decision variable $X$. Now, let us consider this extra PSD constraint $Y \succeq0$ where $$ Y = [Uvec(X)-c]^T\,[Uvec(X)-c] $$ where $U$ is some matrix, $c$ is a vector and $vec(X)$ is the vectorization of $X$. Of course this makes $Y$ a matrix and we require it is PSD.

Questions:

1. What is the Lagrangian of this problem? For each constraint I am supposed to introduce a "Lagrange multiplier" $\ell_i$ which is a dual variable supposedly, so I think I should write something like $$ L = \langle{C,X}\rangle + \langle{\ell_1,X}\rangle + \langle{\ell_2,L}\rangle + \sum_i\lambda_i(\langle A_i,X\rangle-b_i) $$ but I cannot figure out if the term $\langle \ell_2,L\rangle$ makes sense!
2. How can I write down the dual SDP using the Lagrangian? I assume one defines $h := \inf \, L$ and this is equal to $\lambda^Tb $ if $C - \sum_i\lambda_i A_i - \ell_1 -\ell_2 = 0$. Is this correct? Then by eliminating $\ell_1+\ell_2$ one can write down the dual.
3. Can I write down the dual just by looking at the primal without the Lagrangian?

Answer 1

From your definition of $Y$, this matrix is necessarily positive semidefinite. So, the constraint on $Y$ is superfluous.

1. I am not sure to understand what you mean by $\langle \ell_2,L\rangle$ because $L$ is your Lagrangian. But in your case, the Lagrangian is pretty standard as you have a standard SDP in primal form. So, we have that

$$L = \langle{C,X}\rangle - \langle{\ell_1,X}\rangle + \sum_i\lambda_i(\langle A_i,X\rangle-b_i).$$

2. The dual SDP is given by $\max_{\lambda,\ell_1\ge0} \lambda^Tb$ such that $\sum_i\lambda A_i-\ell_1=0$ or, equivalently, $\sum_i\lambda_i A_i\ge0$.

3. Yes, with some habits, you will be able to do that in your head.