Lagrangian Dual Decision Rules for Multistage Stochastic Mixed Integer Programming

Abstract

Multistage stochastic programs can be approximated by restricting policies to follow decision rules. Directly applying this idea to problems with integer decisions is difficult because of the need for decision rules that lead to integral decisions. In this work, we introduce Lagrangian dual decision rules (LDDRs) for multistage stochastic mixed integer programming (MSMIP) which overcome this difficulty by applying decision rules in a Lagrangian dual of the MSMIP. We propose two new bounding techniques based on stagewise (SW) and nonanticipative (NA) Lagrangian duals where the Lagrangian multiplier policies are restricted by LDDRs. We demonstrate how the solutions from these duals can be used to drive primal policies. Our proposal requires fewer assumptions than most existing MSMIP methods. We compare the theoretical strength of the restricted duals and show that the restricted NA dual can provide relaxation bounds at least as good as the ones obtained by the restricted SW dual. In our numerical study, we observe that the proposed LDDR approaches yield significant optimality gap reductions compared to existing general-purpose bounding methods for MSMIP problems.

On the Impact of Deep Learning-based Time-series Forecasts on Multistage Stochastic Programming Policies

“…As previously discussed, there exist many applications that require close coordination of time-series forecasting techniques and MSP. In this work, since our objective is to quantify the performance gap between the combination of general MSP with traditional and modern time-series forecasting techniques, we illustrate the results obtained on a particular MSP application, namely a multi-item lot-sizing problem with backlogging and production lag [15], which contains both discrete and continuous variables. We conduct experiments using two well-known publicly available sales (demand) datasets.…”

“…Regarding the scenario tree-based approaches, there exist some generic decomposition methods [11,28], as well as some algorithms designed for MSPs with binary and continuous variables only [3,41]. For decision rules, the standard approach for the pure continuous case is extended to the binary case via piecewise linear binary functions as the policy form [6], and more recently Lagrangian dual decision rules are proposed for multistage stochastic mixed integer programs [15]. Time-series forecasting methods are designed to take spatio-temporal relations into account.…”

“…Under the common independence assumption, a model is built for each random variable, mostly in the form of a uniform, normal or lognormal distribution. In a more complex setting where the stage-wise independence assumption has been relaxed, previous studies mostly rely on low-order autoregressive models, namely AR(1) and AR (2), to capture some dependencies in uncertain parameters, as in [15,36]. However, the accuracy of the forecasts might have a significant impact on the MSP model results, for both the actual quality of obtained solutions and accuracy of used performance measures, as well as on the number of scenarios required to well express the parameter uncertainty, thus on the complexity of the solution approach.…”

“…Note that in this optimization model, we assume that there is no production related cost. More detailed explanations of the model can be found in [15].…”

“…Regarding the stochastic process {D tj (ξ [t] )} t∈T ,j∈J , for demand scenario generation purposes, we used two publicly available datasets: Walmart Sales Forecasting data [24] and Corporación Favorita Grocery Sales Forecasting data [25]. For the remaining optimization model parameter generation, we followed the steps described in [15] which builds on the steps provided in [22]. We next briefly outline the characteristics of the used demand data.…”