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