Approximate projections for decentralized optimization with SDP constraints

Abstract

We consider distributed convex optimization problems that involve a separable objective function and nontrivial convex local constraints, such as Linear Matrix Inequalities (LMIs). We propose a decentralized, computationally inexpensive algorithm to solve such problems over time-varying directed networks of agents, that is based on the concept of approximate projections. Our algorithm is one of the consensus based methods in that, at every iteration, every agent performs a consensus update of its decision variables followed by an optimization step of its local objective function and local constraint. Unlike other methods, the last step of our method is not a projection to the feasible set, but instead a subgradient step in the direction that minimizes the local constraint violation. We show that the algorithm converges almost surely, i.e., every agent agrees on the same optimal solution, when the objective functions and constraint functions are nondifferentiable and their subgradients are bounded.