In this paper, we introduce a new class of decision rules, referred to as Constant Depth Decision Rules (CDDRs), for multistage optimization under linear constraints with uncertainty-affected right-hand sides. We consider two uncertainty classes: discrete uncertainties which can take at each stage at most a fixed number $d$ of different values, and polytopic uncertainties which, at each stage, are elements of a convex hull of at most $d$ points. Given the depth $\mu$ of the decision rule, the decision at stage $t$ is expressed as the sum of $t$ functions of $\mu$ consecutive values of the underlying uncertain parameters. These functions are arbitrary in the case of discrete uncertainties and are poly-affine in the case of polytopic uncertainties. For these uncertainty classes, we show that when the uncertain right-hand sides of the constraints of the multistage problem are of the same additive structure as the decision rules, these constraints can be reformulated as a system of linear inequality constraints where the numbers of variables and constraints is $O\left(1\right)(n+m)d^{\mu }{N}^2$ with $n$ the maximal dimension of control variables, $m$ the maximal number of inequality constraints at each stage, and $N$ the number of stages.

As an illustration, we discuss an application of the proposed approach to a Multistage Stochastic Program arising in the problem of hydro-thermal production planning with interstage dependent inflows. For problems with a small number of stages, we present the results of a numerical study in which optimal CDDRs show similar performance, in terms of optimization objective, to that of Stochastic Dual Dynamic Programming (SDDP) policies, often at much smaller computational cost.

在本文中，我们引入了一类新的决策规则，称为恒定深度决策规则(CDDR)，用于在右侧受不确定性影响的线性约束下进行多级优化。我们考虑两个不确定性类别：离散不确定性，在每个阶段最多可以采用固定数量$d$ 不同的值，以及多面体的不确定性，在每个阶段，它们是至多凸包的元素 $d$点。鉴于深度 $\mu$ 决策规则，阶段决策 $吨$ 表示为 $吨$ 的功能 $\mu$潜在不确定参数的连续值。这些函数在离散不确定性的情况下是任意的，在多面性不确定性的情况下是多仿射的。对于这些不确定性类别，我们表明，当多阶段问题约束的不确定右侧与决策规则具有相同的加性结构时，这些约束可以重新表述为线性不等式约束系统，其中变量的数量和约束是$哦\left(1\right)(n+米)d^{\mu }{N}^2$ 和 $n$ 控制变量的最大维数， $米$ 每个阶段的不等式约束的最大数量，以及 $N$ 阶段数。

作为说明，我们讨论了所提出的方法在多级随机程序中的应用，该程序出现在具有级间依赖流入的水热生产规划问题中。对于阶段数较少的问题，我们展示了数值研究的结果，其中最佳 CDDR 在优化目标方面表现出与随机双动态规划 (SDDP) 策略相似的性能，通常计算成本要小得多。