On the Absence of Spurious Local Trajectories in Time-varying Nonconvex Optimization

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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.