Lagrangian Dual Decision Rules for Multistage Stochastic Mixed Integer Programming

TL;DR

This paper introduces Lagrangian dual decision rules (LDDRs) for multistage stochastic mixed-integer programming, providing a novel approach to generate bounds and primal policies with fewer assumptions and improved optimality gaps.

Contribution

The work develops LDDRs applied in Lagrangian duals for MSMIP, offering new bounding techniques and primal policy guidance, advancing the solution methodology for integer decision policies.

Findings

LDDR approaches significantly reduce optimality gaps.

Restricted NA duals provide strong relaxation bounds.

Fewer assumptions required than existing methods.

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 on two problem classes, one traditional and one novel, we observe that the proposed LDDR approaches yield significant optimality gap reductions compared to existing general-purpose bounding methods for MSMIP problems.