Aggressive Local Search for Constrained Optimal Control Problems with Many Local Minima

TL;DR

This paper demonstrates that aggressive local search methods can escape local minima in complex constrained optimal control problems with many local minima, potentially converging to a global solution.

Contribution

It introduces an aggressive local search approach with step size adjustments that can jump between connected components, supporting convergence to global optima in non-convex problems.

Findings

Local search can jump between connected components.

Aggressive step size adjustment aids in escaping local minima.

Supports convergence to global solutions despite NP-hardness.

Abstract

This paper is concerned with numerically finding a global solution of constrained optimal control problems with many local minima. The focus is on the optimal decentralized control (ODC) problem, whose feasible set is recently shown to have an exponential number of connected components and consequently an exponential number of local minima. The rich literature of numerical algorithms for nonlinear optimization suggests that if a local search algorithm is initialized in an arbitrary connected component of the feasible set, it would search only within that component and find a stationary point there. This is based on the fact that numerical algorithms are designed to generate a sequence of points (via searching for descent directions and adjusting the step size), whose corresponding continuous path is trapped in a single connected component. In contrast with this perception rooted in convex optimization, we numerically illustrate that local search methods for non-convex constrained optimization can obliviously jump between different connected components to converge to a global minimum, via an aggressive step size adjustment using backtracking and the Armijio rule. To support the observations, we prove that from almost every arbitrary point in any connected component of the feasible set, it is possible to generate a sequence of points using local search to jump to different components and converge to a global solution. However, due to the NP-hardness of the problem, such fine-tuning of the parameters of a local search algorithm may need prior knowledge or be time consuming. This paper offers the first result on escaping non-global local solutions of constrained optimal control problems with complicated feasible sets.