Solution of Probabilistic Constrained Stochastic Programming Problems with Poisson, Binomial and Geometric Random Variables

Probabilistic constrained stochastic programming problems are considered with discrete random variables on the r.h.s. in the stochastic constraints. It is assumed that the random vector has multivariate Poisson, binomial or geometric distribution. We prove a general theorem that implies that in each of the above cases the c.d.f. majorizes the product of the univariate marginal c.d.f’s and then use the latter one in the probabilistic constraints. The new problem is solved in two steps: (1) first we replace the c.d.f’s in the probabilistic constraint by smooth logconcave functions and solve the continuous problem; (2) search for the optimal solution for the case of the discrete random variables. Numerical results are presented and comparison is made with the solution of a problem taken from literature.