# 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.

## 1 Answer

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.