  Choosing decision variables deterministically (deterministic decision-making)
can be regarded as a particular case of choosing decision variables
probabilistically (probabilistic decision-making). It is necessary to
investigate whether probabilistic decision-making can further improve the
expected decision-making performance than deterministic decision-making when
chance constraints exist. The problem formulation of optimizing a probabilistic
decision under chance constraints has not been formally investigated. In this
paper, for the first time, the problem formulation of Chance Constrained
Probability Measure Optimization (CCPMO) is presented to realize optimal
probabilistic decision-making under chance constraints. We first prove the
existence of the optimal solution to CCPMO. It is further shown that there is
an optimal solution of CCPMO with the probability measure concentrated on two
decisions, leading to an equivalently reduced problem of CCPMO. The reduced
problem still has chance constraints due to random variables. We then extend
the sample-based smooth approximation method to solve the reduced problem.
Samples of model uncertainties are used to establish an approximate problem of
the reduced problem. Algorithms for general nonlinear programming problems can
solve the approximate problem. The solution of the approximate problem is an
approximate solution of CCPMO. A numerical example of controlling a quadrotor
in turbulent conditions has been conducted to validate the proposed
probabilistic decision-making under chance constraints.
