How to write lagrangian terms related to only one variable in a semidefinite constraint?

I have a semidefinite problem as follows(which is nonconvex) \begin{alignat}{3} &\min_{x_{un}} \min_{t,H,w} &&t+f( w)\cr &\text{s.t. } &&\begin{bmatrix} K\odot H &1-Xg(w)+ c\\ (1-Xg(w)+c)^\mathsf{T} &t \end{bmatrix} \succeq 0 \cr &&&X=diag([x_k,x_{un}])\in R^{n*n},\cr&&&(w\in R^n, f(w)\in R, g(w)\in R^n,K,H\in R^{n*n}) \end{alignat}

$f(w)$ is a convex quadratic and $g(w)$ is linear function of $w$. I think it's correct to move optimization about $x_{un}$ to most inner problem. Is it correct?

Then how can I write Lagrangian and obtain the dual problem with respect to the most inner variable, i.e, $x_{un}$? My confusion is how to write Lagrangian terms related to two semidefinite constraints, because I don't want to write the Lagrangian terms related to the any other variable.

EDIT: After sometime, I think it may be incorrect to write lagrangian terms related to only single variable. Any hint or comment, Appreciated.

Don't quite understand your question. The primal variable does not matter when you write the Lagrangian form, the constraint matters. For the positive-semidefinite constraint, you also need a matrix as the Lagrangian variable. $$\lambda=\begin{bmatrix} \lambda_{11} & \lambda_{12} \\ \lambda_{21} & \lambda_{22} \\ \end{bmatrix}$$ And when you write the Lagrangian equation, use the trace term for the positive-semidefinite constraint. Suppose your constraint is $A \geq 0$,corresponding Lagrangian term should be: $$Trace(A\lambda)$$ Hope this can help you.