On the Structure of Decision Diagram–Representable Mixed-Integer Programs with Application to Unit Commitment

Salemi, Hosseinali; Davarnia, Danial

doi:10.1287/opre.2022.2353

Cited by 2 publications

(1 citation statement)

References 51 publications

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“…To mitigate this difficulty, restricted/relaxed DDs can be employed inside of the BD framework as demonstrated in Algorithm 1. We refer to this solution method as DD-BD (Salemi and Davarnia 2022a).…”

Section: Continuous Dd Modelsmentioning

confidence: 99%

Solving Unsplittable Network Flow Problems with Decision Diagrams

Salemi¹,

Davarnia²

2023

Preprint

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In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called no-split no-merge requirement arises in unit train scheduling where train consists should remain intact at stations that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network flow problems with standard mixed-integer programming formulations is computationally difficult due to the large number of binary variables needed to determine matching pairs between incoming and outgoing arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based framework that decomposes the underlying two-stage formulation into a master problem that contains the combinatorial requirements, and a subproblem that models a continuous network flow problem. The master problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems output cutting planes that are used to refine the master DD. Computational experiments show a significant improvement in solution time of the DD framework compared with that of standard methods.

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Section: Continuous Dd Modelsmentioning

confidence: 99%

Solving Unsplittable Network Flow Problems with Decision Diagrams

Salemi¹,

Davarnia²

2023

Preprint

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Convexification of Bilinear Terms over Network Polytopes

Khademnia,

Davarnia

2024

Mathematics of OR

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It is well-known that the McCormick relaxation for the bilinear constraint z = xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints [Formula: see text] where xi represents the arc-flow variable in a network polytope, and yj is in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques. Funding: This work was supported by Air Force Office of Scientific Research [Grant FA9550-23-1-0183]; National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2338641]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0001 .

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On the Structure of Decision Diagram–Representable Mixed-Integer Programs with Application to Unit Commitment

Cited by 2 publications

References 51 publications

Solving Unsplittable Network Flow Problems with Decision Diagrams

Solving Unsplittable Network Flow Problems with Decision Diagrams

Convexification of Bilinear Terms over Network Polytopes

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