I have to minimize Trace(C*inv(phi)) with respect to phi. C is a constant matrix. Here phi is positive semidefinite. When I tried this in CVX, it says “Undefined function ‘inv’ for input arguments of type ‘cvx’”. Can we write the objective function in SDP form ?

If `C`

is symmetric positive semidefinite (psd), let `R = chol(C)`

. Then `trace(C*inv(phi) = trace(R'*R*inv(phi)) = trace(R*inv(phi)*R')`

. Then use the solution in Generalizing “trace_inv” for matrix quadratic forms

If C is not psd, then `trace(C*inv(phi))`

is not necessarily convex. For example, in one dimension, let `C = -1`

, then `trace(C*inv(phi)) = -1/phi`

, which is not convex.

If C is invertible and you are willing for `phi*inv(C)`

to be implicitly constrained to be psd, you can simply use `minimize(trace_inv(phi*inv(C)))`