Distributed Gradient Flow: Nonsmoothness, Nonconvexity, and Saddle Point Evasion

The article considers distributed gradient flow (DGF) for multiagent nonconvex optimization. DGF is a continuous-time approximation of descent that often easier to study than its discrete-time counterpart. has two main contributions. First, the optimization nonsmooth, objective functions. It shown converges critical points in this setting. then problem avoiding saddle points. if agents’ functions are assumed be smooth and nonconvex, can only converge point from zero-measure set initial conditions. To establish result, proves stable manifold theorem DGF, which fundamental contribution independent interest. In companion article, analogous results derived algorithms.