Constant Depth Decision Rules for multistage optimization under uncertainty

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 uncertain-ties. 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 con-straints is O(1)(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 Pro-gram 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. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper introduces a new class of decision rules, Constant Depth Decision Rules (CDDRs), for multistage optimization under linear constraints with uncertainty-affected right-hand sides. It demonstrates through mathematical models and application examples the effectiveness of these decision rules in solving complex problems.