Absence of Spurious Local Trajectories in Time-Varying Optimization: A Control-Theoretic Perspective

In this paper, we study the landscape of an optimization problem whose input data vary over time. This time-varying problem consists of infinitely-many individual optimization problems, whose solution is a trajectory over time rather than a single point. To understand when it is possible to find a global solution of a time-varying non-convex optimization problem, we introduce the notion of 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) which, at limit, characterizes the spurious local solutions of the time-varying optimization problem. By building upon this connection, we prove that the absence of spurious local trajectory is closely related to the transient behavior of the proposed ODE. In particular, we show that: (1) if the problem is time-invariant, the spurious local trajectories are ubiquitous since any strict local minimum is a locally stable equilibrium point of the ODE, and (2) if the ODE is time-varying, the data variation may force all ODE trajectories initialized at arbitrary local minima at the initial time to gradually converge to the global solution trajectory. This implies that the natural data variation in the problem may automatically trigger escaping local minima over time.