Gradient and Hessian of Joint Probability Function with Applications on Chance-Constrained Programs

Joint probability function refers to the probability function that requires multiple conditions to satisfy simultaneously.It appears naturally in chanceconstrained programs.In this paper,we derive closed-form expressions of the gradient and Hessian of joint probability functions and develop Monte Carlo estimators of them.We then design a Monte Carlo algorithm,based on these estimators,to solve chance-constrained programs.Our numerical study shows that the algorithm works well,especially only with the gradient estimators.