The Geometry of Equality-constrained Global Consensus Problems

A variety of unconstrained nonconvex optimization problems have been shown to have benign geometric landscapes that satisfy the strict saddle property and have no spurious local minima. We present a general result relating the geometry of an unconstrained centralized problem to its equality-constrained distributed extension. It follows that many global consensus problems inherit the benign geometry of their original centralized counterpart. Taking advantage of this fact, we demonstrate the favorable performance of the Gradient ADMM algorithm on a distributed low-rank matrix approximation problem.