Optimization problems involving sequential decisions in a stochastic environment were studied in Stochastic Programming (SP), Stochastic Optimal Control (SOC) and Markov Decision Processes (MDP). In this paper we mainly concentrate on SP and SOC modelling approaches. In these frameworks there are natural situations when the considered problems are convex. Classical approach to sequential optimization is based on dynamic programming. It has the problem of the so-called ``Curse of Dimensionality", in that its computational complexity increases exponentially with increase of dimension of state variables. Recent progress in solving convex multistage stochastic problems is based on cutting planes approximations of the cost-to-go (value) functions of dynamic programming equations. Cutting planes type algorithms in dynamical settings is one of the main topics of this paper. We also discuss Stochastic Approximation type methods applied to multistage stochastic optimization problems. From the computational complexity point of view, these two types of methods seem to be complimentary to each other. Cutting plane type methods can handle multistage problems with a large number of stages, but a relatively smaller number of state (decision) variables. On the other hand, stochastic approximation type methods can only deal with a small number of stages, but a large number of decision variables.

翻译：随机规划（SP）、随机最优控制（SOC）和马尔可夫决策过程（MDP）研究了随机环境下涉及序贯决策的优化问题。本文主要关注SP和SOC建模方法。在这些框架中，存在所考虑问题为凸的自然情形。序贯优化的经典方法基于动态规划，但其存在所谓的“维数灾难”问题，即计算复杂度随状态变量维数的增加呈指数增长。近年来，求解凸多阶段随机问题的进展基于动态规划方程中待定费用（值）函数的割平面近似。动态环境中的割平面类算法是本文的主要课题之一。我们还讨论了应用于多阶段随机优化问题的随机逼近类方法。从计算复杂度角度来看，这两类方法似乎互为补充：割平面类方法可处理具有大量阶段但状态（决策）变量数相对较少的多阶段问题；而随机逼近类方法仅能处理阶段数较少但决策变量数较大的问题。