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A Fast Distributed Algorithm for Sparse Semidefinite Programs

This paper aims to develop a fast, parallelizable algorithm for an arbitrary decomposable semidefinite program (SDP). To formulate a decomposable SDP, we consider a multi-agent canonical form represented by a graph, where each agent (node) is in charge of computing its corresponding positive semidefinite matrix subject to local equality and inequality constraints as well as coupling and overlapping (consistency) constraints with regards to the agent’s neighbors. Every arbitrary SDP problem can be cast as a decomposable SDP, where the number of nodes and the connectivity of the underlying graph both depend on the sparsity of the original SDP problem. Based on the alternating direction method of multipliers, we design a numerical algorithm, which has a guaranteed convergence under very mild assumptions. Each iteration of this algorithm has a simple closed-form solution, consisting of matrix multiplications and eigenvalue decompositions performed by individual agents as well as information exchanges between neighboring agents. The cheap iterations of the proposed algorithm enable solving large-scale sparse conic optimization problems. The proposed algorithm is applied to several randomly generated SDP problems with over 10 million variables, which are solved in less than 20 minutes.