  Choosing control inputs randomly can result in a reduced expected cost in
optimal control problems with stochastic constraints, such as stochastic model
predictive control (SMPC). We consider a controller with initial randomization,
meaning that the controller randomly chooses from K+1 control sequences at the
beginning (called K-randimization).It is known that, for a finite-state,
finite-action Markov Decision Process (MDP) with K constraints, K-randimization
is sufficient to achieve the minimum cost. We found that the same result holds
for stochastic optimal control problems with continuous state and action
spaces.Furthermore, we show the randomization of control input can result in
reduced cost when the optimization problem is nonconvex, and the cost reduction
is equal to the duality gap. We then provide the necessary and sufficient
conditions for the optimality of a randomized solution, and develop an
efficient solution method based on dual optimization. Furthermore, in a special
case with K=1 such as a joint chance-constrained problem, the dual optimization
can be solved even more efficiently by root finding. Finally, we test the
theories and demonstrate the solution method on multiple practical problems
ranging from path planning to the planning of entry, descent, and landing (EDL)
for future Mars missions.
