  Distributed algorithms for solving coupled semidefinite programs (SDPs)
commonly require many iterations to converge. They also put high computational
demand on the computational agents. In this paper we show that in case the
coupled problem has an inherent tree structure, it is possible to devise an
efficient distributed algorithm for solving such problems. This algorithm can
potentially enjoy the same efficiency as centralized solvers that exploit
sparsity. The proposed algorithm relies on predictor-corrector primal-dual
interior-point methods, where we use a message-passing algorithm to compute the
search directions distributedly. Message-passing here is closely related to
dynamic programming over trees. This allows us to compute the exact search
directions in a finite number of steps. Furthermore this number can be computed
a priori and only depends on the coupling structure of the problem. We use the
proposed algorithm for analyzing robustness of large-scale uncertain systems
distributedly. We test the performance of this algorithm using numerical
examples.
