  In this paper, we show that the bundle method can be applied to solve
semidefinite programming problems with a low rank solution without ever
constructing a full matrix. To accomplish this, we use recent results from
randomly sketching matrix optimization problems and from the analysis of bundle
methods. Under strong duality and strict complementarity of SDP, our algorithm
produces primal and the dual sequences converging in feasibility at a rate of
$\tilde{O}(1/\epsilon)$ and in optimality at a rate of
$\tilde{O}(1/\epsilon^2)$. Moreover, our algorithm outputs a low rank
representation of its approximate solution with distance to the optimal
solution at most $O(\sqrt{\epsilon})$ within $\tilde{O}(1/\epsilon^2)$
iterations.
