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?
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Sign up to join this communityI 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?
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