Constant Depth Decision Rules for multistage optimization under uncertainty

TL;DR

This paper introduces Constant Depth Decision Rules (CDDRs) for multistage optimization under uncertainty, providing a new approach that simplifies complex stochastic problems into linear constraints, with promising computational advantages.

Contribution

The paper proposes a novel class of decision rules, CDDRs, that enable reformulation of multistage stochastic problems with certain uncertainties into linear systems, improving computational efficiency.

Findings

CDDRs can be reformulated as linear inequalities with manageable complexity.

Numerical results show CDDRs perform comparably to SDDP in small-stage problems.

CDDRs offer a computationally efficient alternative for multistage stochastic optimization.

Abstract

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(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 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.