Stochastic Dual Dynamic Programming for Log-linear Autoregressive Uncertainty in the Right-hand Side

We consider the generation of cuts in stochastic dual dynamic programming (SDDP) for multistage stochastic linear programming problems with stagewise dependent uncertainty in the right-hand side described by a nonlinear autoregressive process. We first derive cut formulas for the general nonlinear autoregressive process. For these general cuts, the computational tractability becomes a major challenge for SDDP-type algorithms. We are, however, able to develop tractable closed-form cut formulas for the special case of log-linear autoregressive processes. The obtained cuts are in general non-convex in the history of the stochastic process, but still linear in all decision variables, and thus can be directly incorporated into the subproblems in SDDP without compromising their linearity. If solvers do not allow for this, our formulas can be used to adapt the intercept of a given cut to a scenario at hand in a computationally tractable way. In this sense, cuts generated for one specific scenario are shared with other scenarios. Our findings are supported by a computational study of a hydrothermal scheduling problem, which demonstrates a favorable performance of our proposed version of SDDP compared to approaches from the literature.