I am struggling to solve an optimization problem of the following form:


$$\begin{array}{ll} \underset{A}{\text{maximize}} & \log \det (A) \\ \text{subject to} & a^T A^{-1} a \le b\end{array}$$


Is there any solution for it?



Optimization problem involving matrix
I will presume you want $A$ to be constrained to be symmetric (hermitian) psd. In that case, this is a convex optimization problem which is a Linear Semidefinite Programming problem (SDP) a.k.a. Linear Matrix Inequality (LMI).


A convex optimization modeling tool, such as CVX, can formulate this as a standard Linear SDP and call a solver to solve it.


Here is the code for CVX.



```
cvx_begin
variable A(n,n) hermitian semidefinite
maximize(log_det(A))
subject to
matrix_frac(a,A) <= b
cvx_end

```

