Programming Under Uncertainty: The Equivalent Convex Program

This paper is an attempt to describe and characterize the equivalent convex program of a two-stage linear program under uncertainty. The study has been divided into two parts. In the first one, we examine the properties of the solution set of the problem and derive explicit expressions for some particular cases. The second section is devoted to the derivation of the objective function of the equivalent convex program. We show that it is convex and continuous. We also give a necessary condition for its differentiability and establish necessary and sufficient conditions for the solvability of the problem. Finally, we give the equivalent convex program of certain classes of programming under uncertainty problems, i.e., when the constraints and the probability space have particular structures.