Measures as Lagrange Multipliers in Multistage Stochastic Programming.

摘要

A duality theory is developed for multistage convex stochastic programming problems whose decision (or recourse) functions can be approximated by continuous functions satisfying the same constraints. Necessary and sufficient conditions for optimality are obtained in terms of the existence of multipliers in the class of regular Borel measures on the underlying probability space, these being decomposable, of course, into absolutely continuous and singular components with respect to the given probability measure. This provides an alternative to the approach where the multipliers are elements of the dual of L(infinity) with an analogous decomposition. However, besides the existence of strictly feasible solutions, special regularity conditions are required, such as the laminarity of the probability measure, a property introduced in an earlier paper. These are crucial in ensuring that the minimum in the optimization problem can indeed be approached by continuous functions. (Author)