optimization related to sdp I have the dual solution of an sdp problem and strong duality hold in this case, I have the dual feasible solutions . From the Dual feasible solutions can i get the primal feasible solution? 
 A: I'll assume in my answer that you're using the convention that the primal problem is:
$\max tr(CX) $
subject to
$tr(A_{i}X)=b_{i}\;\; i=1, 2, ..., m$ 
$X \succeq 0$
where $X$ is an $n$ by $n$ symmetric and PD matrix.  The dual is 
$\min b^{T}y$
subject to 
$A^{T}(y)-C=Z$
$Z \succeq 0$
The same basic approach carries over to other primal-dual formulations of the SDP.  
The KKT conditions for the primal dual pair are that
$A(X)=b$
$A^{T}(y)-C=Z$
$XZ=0$
$X \succeq 0$
$Z \succeq 0$
The complementary condition $XZ=0$, together with the requirement that $X$ and $Z$ must be PSD lead to:


*

*$Z$ can be written as $Z=Q \Lambda Q^{T}$ where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with $\Lambda_{1,1} \geq \Lambda_{2,2} \geq ... \geq \Lambda_{k,k} \geq \Lambda_{k+1,k+1}=...=\Lambda_{n,n}=0$.

*$X$ can be written as $X=QWQ^{T}$ where $W$ is also diagonal with 
$W_{1,1}=W_{2,2}=...=W_{k,k}=0$, and $W_{j,j} \geq 0$ for $j=k+1, k+2, ..., n$.  
So, you can (in theory) find an optimal $X$ from an optimal $y$ and $Z$ by solving the equations 
$tr(A_{i}X)=b_{i}\;\; i=1, 2, ..., m$
$X=QWQ^{T}$
where $W$ is a diagonal matrix matrix satisfying the requirements above.  
This is typically severely over determined.  In practice, you'll need to find a solution that satisfies the $tr(A_{i}X)=b_{i}$ constraints well enough in the least squares sense.  This simplifies down to a nonnegative linear least squares problem in the vector of variables $W_{k+1,k+1}$, $W_{k+2,k+2}$, $...$, $W_{n,n}$.  
In practice, unless you have an extremely good dual solution, it's unlikely that this approach will yield a very good primal solution- this has been a persistent problem with various dual methods for semidefinite programming such as the spectral bundle method or the dual interior point method implemented in DSDP.  Primal-dual interior point methods don't have this problem, but of course these methods don't scale well to larger problem instances.  
