Programming under Uncertainty: The Complete Problem

摘要

We define the complete problem of a two-stage linear programming under uncertainty, to be: Minimize z(x) = E sub xi ( c x + q(+)y(+) + q(-)y(-)) subject to A x = b; T x + I y(+) + I y(-) = xi x > or = O y(+) > or = y(-) > or = O where x is the first-stage decision variable, the pair (y(+), y(-)) represents the second-stage decision variables. In order to solve this class of problem, we derive a convex programming problem, whose set of optimal solutions is identical to the set of optimal solutions of our original problem. This problem is called the equivalent convex programming. If the random variable xi has a continuous distribution, we give an algorithm to solve the equivalent convex program. Moreover, we derive explicitly the equivalent convex program for a few common distributions. (Author)