Could you help me formulate the dual problem to this SDP?
maximize $\frac{1}{2} Tr(GW)$,
subject to $ G \ge 0$ (and G symmetric), and $ \forall i$, $ G_{ii} = G_{1i} = G_{i1} $
Note that $G$ and $W$ are $9\times 9$ matrices. The procedure I have used so far is to write the Lagrangian, then from it the dual function and the dual problem, but I am not sure I am doing these steps correctly.
Thank you for your help.
I am a beginner in optimization theory, So wait for someone to confirm this.
My idea is to start with finding the standard form of an SDP, witch is :
$$\begin{array}{rll} {\displaystyle\min_{X \in \mathbb{S}^n}} & \langle C, X \rangle_{\mathbb{S}^n} & \\ \text{subject to} & \langle A_i, X \rangle_{\mathbb{S}^n} = b_i, \quad i = 1,\ldots,m & (P) \\ & X \succeq 0 & \end{array}$$
To do that, I am thinking to use the same strategy as in my answer of this question I posed ( dual problem of a Semidefinite programming in a non-standard forme)
Let $ E^{(k, l)} \in S^ {9\times 9} $ ($k,l=1,..., 9)$ denote the symmetric matrices with $(k, l)$th entry = $(l, k)$th entry $= 1$ and zero elsewhere. When $k=l$, $ E^{(k, k)} $ denotes the matrix with ($k,k)$th entry 1 and all other elements 0. Clearly, we have $E^{(l,k)}=E^{(k,l)}$ for any $(k,l).$ Note that for any matrix $G=(g_{i,j}) \in S^{9\times 9},$ it can be represented as $G=\sum_{k=1}^{9} \sum_{l=k}^{9} g_{k,l}E^{(k,l)},$ and $ \langle E^{(k, l)}, G \rangle = g_{k,l}+g_{l,k} = 2g_{k,l}$ for $k\not=l,$ and $ \langle E^{(k, k)}, G \rangle = g_{k,k} $.
your problem can be written in the form :
$$ \begin{eqnarray} & \min & \left\langle \frac{1}{2} W, G \right\rangle \\ & \textrm{s.t.} & \langle 2E^{(1,1)}-E^{(1,i)}, G \rangle =0, ~i=2,..., 9, \\ & & \langle E^{(1,1)}-E^{(i, i)}, G \rangle =0, ~i=2,..., 9, \\ & & G \succeq 0 \end{eqnarray} $$
The dual of this problem is well known.
