$$\underset{x}{minimize} \frac{1}{2}x^TP_{0}x + q_{0}^{T}x$$ $$subject \; to \quad \frac{1}{2}x^TP_{i}x + q_{i}^{T}x + r_{i}\leq 0\quad i \in \{1,2,\ldots,m\}$$ $$\quad Ax = b$$
It seems to me that this type of program is specifically well suited for a Sequential Quadratic Programming (SQP) solution, especially when the $P_{i}$ are positive semi-definite. However, I have only seen SDP relaxations in literature.
• If $P_i$ are positive semidefinite, you would not use a semidefinite relaxation, but simply solve the problem using a solver for convex quadratically constrained programs (special case of SOCP for which there are several solvers). And in the nonconvex case, why would SQP be any better than any other method to general nonconvex nonlinear programs? If you use e.g. fmincon in MATLAB, you can select the algorithm to be SQP if you wish. Same with many other solvers I guess. – Johan Löfberg Feb 9 '18 at 11:46