There seem to be a lot of work on finding the nearest positive semidefinite matrix to a real symmetric matrix, but other than via an eigendecomposition, which is slow (as below), does anyone know of a way to find the nearest PSD matrix to a complex Hermitian matrix?
# take the eigendecomposition, set any negative
# eigenvalues to 0, and reconstruct
D,V = eigen(Hermitian(H))
D .= max.(D,0)
Try formulating your problem as a conic program and use a conic solver, e.g. GitHub - mariohsouto/ProxSDP.jl: Semidefinite programming optimization solver. Also if you ask in the optimization category, you may find more helpful comments from the SDP folks there.
I don’t know much about this type of problems, but often methods that work or symmetric matrices work for hermitian as well, by changing transposes to adjoints and adding a few