That’s my point: submitting the problem with a 0 objective is just telling the program: once you found an initial solution (i.e., once you’ve solved the “feasibility problem”), stop, and report that solution.
Now, EITHER there is a way to phrase the feasibility problem as an SDP with an obvious starting value (which I don’t think is the case). In which case the question would be: what is the formulation of the SDP?
OR, the algorithm for finding the initial solution (i.e., for solving the “feasibility problem”) is different from the one you use to solve the SDP once you have an initial solution. In that case, the question (the right one, I think) is: what is the algorithm used to find a feasible solution?
Is that algorithm different if you just want a feasible solution, not necessarily a “good” starting value for the optimization problem??