We propose a new algorithm for solving multistage stochastic mixed integer linear programming (MILP) problems with complete continuous recourse. In a similar way to cutting plane methods, we construct nonlinear Lipschitz cuts to build lower approximations for the non-convex cost to go functions. An example of such a class of cuts are those derived using Augmented Lagrangian Duality for MILPs. The family of Lipschitz cuts we use is MILP representable, so that the introduction of these cuts does not change the class of the original stochastic optimization problem. We illustrate the application of this algorithm on two simple case studies, comparing our approach with the convex relaxation of the problems, for which we can apply SDDP, and for a discretized approximation, applying SDDiP.

中文翻译：

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随机 Lipschitz 动态规划

我们提出了一种新算法，用于解决具有完全连续资源的多级随机混合整数线性规划 (MILP) 问题。以与切割平面方法类似的方式，我们构造非线性 Lipschitz 切割来构建非凸成本函数的较低近似值。此类切割的一个示例是使用 MILP 的增广拉格朗日对偶导出的那些切割。我们使用的 Lipschitz 切割系列是 MILP 可表示的，因此这些切割的引入不会改变原始随机优化问题的类别。我们在两个简单的案例研究中说明了该算法的应用，将我们的方法与问题的凸松弛进行了比较，我们可以应用 SDDP，而对于离散化近似，应用 SDDiP。