  Semidefinite programs (SDP) are one of the most versatile frameworks in
numerical optimization, serving as generalizations of many conic programs and
as relaxations of NP-hard combinatorial problems. Their main drawback is their
computational and memory complexity, which sets a practical limit to the size
of problems solvable by off-the-shelf SDP solvers. To circumvent this fact,
many algorithms have been proposed to exploit the structure of particular
problems and increase the scalability of SDPs for those problem instances.
Progress has been less steep, however, for general-case SDPs. In this paper,
motivated by earlier results by Ahmadi and Hall, we show that a general SDP can
be solved to $\epsilon$-optimality, in polynomial time, by performing a
sequence of less computationally demanding Linear or Second Order Cone
programs. In addition, we provide a bound on the number of iterations required
to achieve $\epsilon$-optimality. These results are illustrated using random
SDPs and well-known problems from the SDPLib dataset.
