Representing $\mathbf{Tr}(A + BC^{-1}B^T)< K$ as an LMI

Question

I am trying to implement a convex optimization and I have the constraint $\mathbf{Tr}(A + BC^{-1}B^T)< K$ where $A\succeq0,B,C\succ0$ are the decision variables. Can it be transformed into a linear constraint or a linear matrix inequality?

Answer 1

The trace can also be written as

$$ \mathbf{Tr}(A + B\,C^{-1}B^\top) = \sum_{i=1}^n e_i^\top (A + B\,C^{-1}B^\top)\,e_i, \tag{1} $$

with $n$ such that $A \in \mathbb{R}^{n\times n}$ and $e_i$ the $i$th column of an $n \times n$ identity matrix. By introducing intermediate scalar variables $\alpha_i$ one can write the initial inequality in an indirect way by using

\begin{align} e_i^\top (A + B\,C^{-1}B^\top)\,e_i &< \alpha_i, \forall\,i = 1 \dots n, \tag{2} \\ \sum_{i=1}^n \alpha_i &< K. \tag{3} \end{align}

The inequality from $(3)$ is already a linear inequality. The inequality from $(2)$ can be formulated as linear (matrix) inequality by using the Schur complement. It can be noted that applying the Schur complement to each inequality from $(2)$ does require the additional assumptions that $C$ is positive definite.