Hello I have a question on the dual variable in SDP problem
Actually, I'm reading the papers about QCR ( reformulating the 0-1 QCQP problem into a convex problem with tightest bound), but I will skip the detail and ask you the one that I haven't understood.
Here is a SDP problem (denote by (SDP) )
$$ max\ r \\ s.t $$
$$ \left[ \begin{matrix} -r & \frac{1}{2}(c+u)^T \\ \frac{1}{2}(c+u) & Q-diag(u) \\ \end{matrix} \right] \ge 0 $$ $$ where \ r \in R \ \ \ u \in R^n $$
Then the (SDP) is a convex problem and satisfies Slater's condition.(take large negative r etc) Thus it has a strong duality property. The dual of (SDP) is as following(denote by (DSDP)
$$ min \ c^tx + Tr(QX) \\ s.t\ \ X_{ii} = x_i \ \ \ \ (1)\\ \left[ \begin{matrix} 1 & x^T \\ x & X \\ \end{matrix} \right] \ge 0\\ where\ x \in R^n \ \ X \in R^{n*n} $$ where Tr is a trace of matrix.
Here is my question. I understand the optimal value of (SDP) and (DSDP) are same due to the strong duality. However, many papers say that the optimal value of u in SDP are given by the optimal values of the dual variables associated with (1) in DSDP. Do you have an idea on why the optimal value of u in primal problem(SDP) can be obtained by dual variable of dual problem(DSDP)?
I heard the dual of dual problem is same with the primal problem in LP thus I take the dual problem of (DSDP), but it is not same with (SDP).
ref) Using a mixed integer quadratic programming solver for the unconstrained quadratic 0-1 program Improving the performance of standard solvers for quadratic 0-1 programs by a tight convex reformulation: The QCR method