In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP). By writing Dynamic Programming equations for the dual, we can employ an SDDP type method, called Dual SDDP, which solves these Dynamic Programming equations. allows us to compute a sequence of nonincreasing deterministic upper bounds for the optimal value of the problem. Since the Relatively Complete Recourse (RCR) condition may fail to hold for the dual (even for simple problems), we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts, that converge to the optimal value of the dual (and therefore primal when there is no duality gap) problem under mild assumptions. We also show that optimal dual solutions can be obtained using dual information from Primal SDDP (applied to the original primal MSLP) subproblems.

As a byproduct of the developed methodology we study sensitivity of the optimal value of the problem as a function of the involved parameters. For the sensitivity analysis we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them, i.e., we need to solve the dual problem.

Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.

在本文中，我们研究了多级随机线性规划 (MSLP) 的对偶性。通过为对偶编写动态规划方程，我们可以采用称为对偶 SDDP 的 SDDP 类型方法来求解这些动态规划方程。允许我们为问题的最优值计算一系列非增加的确定性上限。由于相对完全资源 (RCR) 条件可能无法对对偶（即使对于简单问题）成立，我们设计了 Dual SDDP 的两个变体，即带有惩罚的 Dual SDDP 和带有可行性削减的 Dual SDDP，收敛到最优值温和假设下的对偶（因此在没有对偶间隙时是原始的）问题。

作为开发方法的副产品，我们研究了问题最优值的敏感性作为所涉及参数的函数。对于敏感性分析，我们提供了关于参数的价值函数导数的公式，并说明了它们在库存问题上的应用。由于这些公式涉及最优对偶解，我们需要一种算法来计算这些解以使用它们，即，我们需要解决对偶问题。

最后，作为所开发工具的概念证明，我们展示了计算库存问题最优值的敏感性的数值实验结果作为需求过程参数的函数，并比较了库存上的 Primal 和 Dual SDDP 以及水热规划问题。