  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.
