We consider the following semidefinite program where the variables are semidefinite matrices, i.e., their set is $S_n^+$.We want to solve the following problem
$$\inf_{X \in M_3(\mathbb{R})} X_{3,3} + \iota_{S_3^+}(X)$$
under the constraints :
$$X_{1,2} + X_{2,1} + X_{3,3} = 1, \qquad X_{2,2} = 0$$
I solved the primal problem and I want to compute the dual of this problem. I wrote the Lagrangian
$$L(X,\lambda, \mu) = X_{3,3} + \lambda^T(X_{1,2} + X_{2,1} + X_{3,3} - 1) + \mu^T(X_{2,2})$$
The dual problem is
$$\min_{x \in S_3^+} X_{3,3} + \lambda(X_{1,2} + X_{2,1} + X_{3,3} - 1) + \mu(X_{2,2})$$
Then I am blocked to find the solution of the dual problem. I think that if $X_{1,2} + X_{2,1} + X_{3,3} = 0$ and $X_{2,2} = 0$, then the dual problem is
$$\min_{\lambda \in \mathbb{R}^+} -\lambda$$ Any help, please?
EDIT
In fact I think I made a mistake. The dual solution may be $- \infty$
I am now blocked by this question
We define $f$ by $f(x) = <X,E_{3,3}> + \iota_{S_3^+}$ with $E_{3,3} = diag(0,0,1)$.
And $G(X) = [X_{1,2} + X_{2,1} + X_{3,3}; X_{2,2}]$ $e_1 = [1;0]$ and $A(X) = G(X) - e_1$
I have to show that $[0; \varepsilon] \in A(dom(f)$ if and only if $\varepsilon \geq 0$. It is OK. But I do not manage to know why then the constraints are not qualified.
EDIT $\iota_{S_3^+}$ is the function which is $\infty$ if its argument does not belong to $S_3^+$ and $0$ otherwise.