I’m trying to optimize a function with respect to a real matrix M with all eigenvalues real and lying in the open interval (0,1).
I know that for a real symmetric positive-definite matrix S, we can use the Cholesky factorization S = L L^T where elements of L are free parameters.
Is there a similar way of enforcing the above eigenvalue constraint?
Use semidefinite programming with the following constraints:
- S \succcurlyeq 0
- T = I - S
- T \succcurlyeq 0
where I is the identity matrix and S and T are matrix decision variables.
Thank you for your reply. So to optimize over matrices M which satisfy my eigenvalue constraint, I should equivalently optimize over general positive (semi-) definite matrices T and S, subject to the linear constraint that T=I-S?