  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.
