Biased Kernel Density Estimators for Chance Constrained Optimal Control Problems

A method is developed for transforming chance constrained optimization problems to a form numerically solvable. The transformation is accomplished by reformulating the chance constraints as nonlinear constraints using a method that combines the previously developed Split-Bernstein approximation and kernel density estimator (KDE) methods. The Split-Bernstein approximation in a particular form is a biased kernel density estimator. The bias of this kernel leads to a nonlinear approximation that does not violate the bounds of the original chance constraint. The method of applying biased KDEs to reformulate chance constraints as nonlinear constraints transforms the chance constrained optimization problem to a deterministic optimization problems that retains key properties of the chance constrained optimization problem and can be solved numerically. This method can be applied to chance constrained optimal control problems. As a result, the Split-Bernstein and Gaussian kernels are applied to a chance constrained optimal control problem and the results are compared.