  A warm start method is developed for efficiently solving complex chance
constrained optimal control problems. The warm start method addresses the
computational challenges of solving chance constrained optimal control problems
using biased kernel density estimators and Legendre-Gauss-Radau collocation
with an $hp$ adaptive mesh refinement method. To address the computational
challenges, the warm start method improves both the starting point for the
chance constrained optimal control problem, as well as the efficiency of
cycling through mesh refinement iterations. The improvement is accomplished by
tuning a parameter of the kernel density estimator, as well as implementing a
kernel switch as part of the solution process. Additionally, the number of
samples for the biased kernel density estimator is set to incrementally
increase through a series of mesh refinement iterations. Thus, the warm start
method is a combination of tuning a parameter, a kernel switch, and an
incremental increase in sample size. This warm start method is successfully
applied to solve two challenging chance constrained optimal control problems in
a computationally efficient manner using biased kernel density estimators and
Legendre-Gauss-Radau collocation.
