Some Classes of Valid Inequalities and Convex Hull Characterizations for Dynamic Fixed-Charge Problems under Nested Constraints

The authors also gratefully acknowledge the detailed comments of two anonymous referees regarding relationships with lot-sizing problems. ii Abstract This paper studies the polyhedral structure of dynamic fixed-charge problems that have nested relationships constraining the flow or activity variables. Constraints of this type might typically arise in hierarchical or multi-period models, but might also be induced among choices of key variables via an LP-based post-optimality analysis. In particular, this structure is also inherent in capacitated lot-sizing problems. We characterize several classes of valid inequalities and inductively derive convex hull representations in a higher dimensional space for such dynamic fixed-charge problems having nested constraints using lifting constructs based on the Reformulation-Linearization Technique (RLT). Relationships with certain known classes of valid inequalities for single item capacitated lot-sizing problems are also identified. This offers insights into how the RLT could be used as a unifying mechanism for generating such classes of valid inequalities, and prompts several research avenues that we propose for future investigation.