maximize log determinant subject to a linear constraint Does anyone know any efficient method to solve the following problem?
$ (\alpha,\beta) = \text{argmax} \log \det (\alpha K_1 + \beta K_2)$
s.t. $c_1 \alpha + c_2 \beta = c_3, \alpha\geq0, \beta\geq 0$
where $K_1$ and $K_2$ are known positive semi-definite matrix and $c_1$, $c_2$ and $c_3$ are known constant.
Many thanks!

I guess you can use below property:
The determinant behaves like a linear function on the rows of the matrix such as
$$ det(\alpha K + \beta L)=\begin{vmatrix} \alpha K_{1,1}+\beta L_{1,1}&\alpha K_{1,2}+\beta L_{1,2}\\\alpha K_{2,1}+\beta L_{2,1}&\alpha K_{2,1}+\beta L_{2,2}\end{vmatrix}$$
$$\Rightarrow \alpha^2\begin{vmatrix} K_{1,1}&K_{1,2}\\K_{2,1} & K_{2,1}\end{vmatrix}+\alpha \beta\begin{vmatrix} K_{1,1}&K_{1,2}\\L_{2,1} & L_{2,1}\end{vmatrix}+\alpha \beta\begin{vmatrix} L_{1,1}&L_{1,2}\\K_{2,1} & K_{2,1}\end{vmatrix}+\beta^2\begin{vmatrix} L_{1,1}&L_{1,2}\\L_{2,1} & L_{2,1}\end{vmatrix}$$
It may be easier to construct the matrices than taking inverses.
