A weak duality theorem for stochastic linear programming

Abstract

A linear programming problem is said to be stochastic if one or more of the coefficients in the objective function or the system of constraints or resource availabilities is known only by its probability distribution. A distinction is usually made between two related approaches to stochastic linear programming, the active and passive approach respectively. An extension of the duality theorem of non-stochastic or deterministic programming problem has been attempted in this paper in the area of stochastic linear programming in its two approaches. The method of proof is based on the idea that since the parameter space defined by a stochastic linear programme is the topological product of the real line with itself, it forms a first countable topological space. Using a set of distinct and selected points in the parameter space the concepts of feasibility, optimality and duality are extended to stochastic linear programming problems of arbitrary dimensionality. Based on the non-singular regions of the parameter space of a stochastic linear programming problem the theorem utilizes the conditions of convergence of the sequence of distinct and selected points in the parameter space to a limit point and thereby generalizes the duality theorem in the stochastic case. Furthermore it is shown that the regions of feasibility of the active and passive approaches of stochastic linear programming may be different, so that on this basis it may be possible to establish some inequality relations for the optimal solutions defined for the respective feasible regions.