I'm having a problem to find the Dual of a Semidefinite programing problem:
$$\min\;\;(tr(U)+tr(V))/2$$
$$s.t.\;\; \left[ \begin{array}{cc} U & X \\ X^T & V \end{array} \right]\succeq0$$
$$X_{ij}=M_{ij}\;\;(i,j)\in\Omega$$
Where $tr()$ is the trace operator, $U, V$ and $X$ are the matrix variables of the problem and $M$ is a given matrix.
It is known that a general SDP has the followind Primal Dual pair:
$$P) \;\; \min \;\; tr(C^TX)$$
$$s.t.\;\; tr(A_{i}^TX)=b_i\;\;i=1,...,m$$ $$X\succeq 0$$
$$D)\;\; \max b^Ty$$
$$s.t.\;\; \sum_{i=1}^{m}A_iy_i + S = C $$ $$S\succeq0$$
But i can't find the way to modify my problem to this form.
You just need to transform everything into the standard form. Writing the primal problem in the standard form is easy. In fact, we can set
$$ C = \left[ \begin{array}{cc} I/2 & 0 \\ 0 & I/2 \end{array} \right], A_k = \left[ \begin{array}{cc} 0 & 1_{ij}/2 \\ 1_{ji}/2 & 0 \end{array} \right], b_k = M_{ij}, \forall (i,j) \in \Omega.$$
Therefore, the dual will be $$ \max ~ tr(Y, M) $$ $$ s.t. ~ \left[ \begin{array}{cc} I/2 & -Y/2 \\ -Y^\top/2 & I/2 \end{array} \right] \succeq 0$$ where $M$ is a matrix whose elements are zero if $(i,j) \notin \Omega$ otherwise $M_{ij}$.