Strong duality of SDPs

Question

On pp. 654 of Boyd's book, it is claimed that strong duality holds between the SDPs B.2 and B.3 (at the bottom of this page). Does it require additional assumption that one of them is strictly feasible (or it is already implied in the problem)? Thanks.

Answer 1

Lagrangian for B.2, is $L(\lambda,\gamma,\eta,\Big[\begin{matrix}X &x\\x^T&y \end{matrix}\Big])=-\gamma-<\Big[\begin{matrix}X &x\\x^T&y \end{matrix}\Big],\Big[\begin{matrix}A_0+\lambda*A_1 &b_0+\lambda b_1\\(b_0+\lambda b_1)^T&c_0+\lambda c_1-\gamma \end{matrix}\Big]>-\eta \lambda$

or

$L(\lambda,\gamma,\eta,\Big[\begin{matrix}X &x\\x^T&y \end{matrix}\Big])=-\gamma(y-1)-\Big(<A_0,X>+2 b_0^T x +c_0\Big)-\Big(<A_1,X>+2 b_1^T x + c_1+\eta \Big)\lambda$

In order for this problem to be feasible and bounded, we must have

$<A_1,X>+2 b_1^T x + c_1+\eta =0$ and $y-1=0$.

And you reach the dual.