  In this paper, we study the landscape of an online nonconvex optimization
problem, for which the input data vary over time and the solution is a
trajectory rather than a single point. To understand the complexity of finding
a global solution of this problem, we introduce the notion of \textit{spurious
(i.e., non-global) local trajectory} as a generalization to the notion of
spurious local solution in nonconvex (time-invariant) optimization. We develop
an ordinary differential equation (ODE) associated with a time-varying
nonlinear dynamical system which, at limit, characterizes the spurious local
solutions of the time-varying optimization problem. We prove that the absence
of spurious local trajectory is closely related to the transient behavior of
the developed system. In particular, we show that if the problem is
time-varying, the data variation may force all of the ODE trajectories
initialized at arbitrary local minima at the initial time to gradually converge
to the global solution trajectory. We study the Jacobian of the dynamical
system along a local minimum trajectory and show how its eigenvalues are
manipulated by the natural data variation in the problem, which may
consequently trigger escaping poor local minima over time.
