# Dual of semidefinite program (SDP)

I want to derive the dual of the following SDP

$$\begin{array}{ll} \displaystyle\min_{\substack{W_{1} \in \mathbb{S}^{m}, W_{2} \in \mathbb{S}^{n}}} & \frac 12 \mbox{tr}(W_{1}) + \frac 12 \mbox{tr}(W_{2}) + \gamma \, \mbox{tr}(e^T e Z)\ \\ \text{subject to} & \begin{bmatrix} W_{1} & X \\ X^T & W_{2} \end{bmatrix} \succeq 0 \end{array}$$ $$\begin{array}{ll} -Z_{ij}\leq Y_{ij} \leq Z_{ij}\\ e^T X e = mn\\ X_{ij}-Y_{ij}=0\\ X_{ij}\leq 1\ \end{array}$$

I feel confused about the last term of the objective function and some constraints. Thank you.

• Are $W_i$ the only variables? What is the dual you can come up with? – LinAlg Apr 3 '18 at 18:53
• I think you probably intend to have $ee^T$ inside of that objective term instead of $ee^T$. – Michael Grant Apr 4 '18 at 0:13