A new method to solve stochastic programming problems under probabilistic constraint with discrete random variables

In this dissertation, probabilistic constrained stochastic programming problems are considered with discrete random variables on the r.h.s. in the stochastic constraints. In Chapter 2 and 3, 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. In Chapter 4, numerical examples are presented and comparison is made with the solution of a problem taken from the literature. In Chapter 5, some properties of p level efficient points of a random variable are studied, and a new algorithm to enumerate all the p level efficient points is developed. In Chapter 6, p level efficient points in linear systems are studied.