Properties and Solutions of a Class of Stochastic Programming Problems with Probabilistic Constraints

OF THE DISSERTATION Properties and solutions of a class of stochastic programming problems with probabilistic constraints by Kunikazu Yoda Dissertation Director: András Prékopa We consider two types of probabilistic constrained stochastic linear programming problems and one probability bounding problem. The first type involves a random left-hand side matrix whose rows are independent and normally distributed. The quasi-concavity of the constraining function needed for the convexity of the problem is ensured if the factors of the function are uniformly quasi-concave. A necessary and sufficient condition is given for that property to hold. We show practical application in optimal portfolio construction. The second type is the stochastic multidimensional knapsack problem which involves a random left-hand side matrix with independent components and 0-1 decision variables. We show that the problem is convex, under some condition on the parameters, for special continuous and discrete distributions: gamma, normal, Poisson, and binomial. Numerical experiments suggest that the problem can be solved as efficiently as its deterministic version for moderate sized instances. In the last problem, we formulate the linear programming problems that give improved lower and upper bounds on the probability of the union of events when the probabilities of some individual or intersections of events in a first few terms of the inclusion-exclusion principle are 0 or very small.