Solution theorems in probabilistic programming: A linear programming approach

Abstract

Abstract : For some years research on solution theorems in probabilistic programming has been dormant. The obvious consequences of formal similarities to deterministic mathematical programming problems had been rapidly exhausted by researchers. Currently, however, the deeper study which was taking place during the 'dormant period' has begun to produce results. On the one hand theorems characterizing optimal classes of stochastic decision rules for various general change-constrained problems have been obtained. On the other hand, a great amount of effort has been expended on the special class of problems called linear programming problems under uncertainty, usually 2-stage and under still more special assumptions. The general objective of these specializations has been to attain results and thereby to gain insight and technique to reapproach more fruitfully the more important and general but more recondite probabilistic programming problems. To this end, few abstractions or devices, from finite-dimensional Banach spaces to the Kakutani fixed-point theorem appear to have gone untried, except, perhaps, the ancient one of study and correlation of the existent results of other researchers. It is the purpose of this paper to provide some such correlation and a redirection so that these simpler probabilistic programming problems may be overcome in all generality with new, simpler methods which offer some promise of extension to the more involved chance constrained (and other probabilistic) models.