As I know, the constraints and objective function are all linear functions in the conventional SDP problem.

And there exist some solvers which can solve the nonlinear SDP problems.

There exist three constraint as follows:

a>=0;

inv_pos(a)<=trace(W),

W>=0;

where ‘a’ and ‘W’ are two variables, the first one is a scalar, the second one is the Hermitian complex variable.

We point out that when a>=0, the function ‘inv_pos(a)’ is convex.

Therefore, we can use the first-oder Taylor expansion to reformulate the second constraint.

Question:

Besides the Taylor expansion, how can we deal with the above constraint? or without dealing with the above constraint, we can use some special solvers to solve the cosrresponding SDP problem directly?