Valid Cuts for the Design of Potential-based Flow Networks
Pascal Börner, Max Klimm, Annette Lutz, Marc E. Pfetsch, Martin Skutella, Lea Strubberg
TL;DR
This work tackles the problem of designing cost-efficient networks for flows governed by a potential-based physics model, formulated as a challenging MINLP with binary arc choices and nonlinear flow constraints. The authors develop a novel, polynomial-time separable family of valid inequalities for fractional relaxations, enabling a branch-and-cut approach that tightens the relaxations and guides the search. They extend these inequalities from simple multi-path and s-t flows to general b-transshipments and, crucially, to the topology-optimization setting by introducing an induced network N^x and proving polynomial-time separability via disjoint cuts and submodular optimization. Computational experiments on real-world gas networks demonstrate meaningful performance improvements, showing the practical potential of the approach for infrastructure design under physically grounded flow models. Overall, the paper provides a principled, scalable method to integrate physics-based flow constraints into combinatorial network design through cut-based inequalities and efficient separation routines.
Abstract
The construction of a cost minimal network for flows obeying physical laws is an important problem for the design of electricity, water, hydrogen, and natural gas infrastructures. We formulate this problem as a mixed-integer non-linear program with potential-based flows. The non-convexity of the constraints stemming from the potential-based flow model together with the binary variables indicating the decision to build a connection make these programs challenging to solve. We develop a novel class of valid inequalities on the fractional relaxations of the binary variables. Further, we show that this class of inequalities can be separated in polynomial time for solutions to a fractional relaxation. This makes it possible to incorporate these inequalities into a branch-and-cut framework. The advantage of these inequalities is lastly demonstrated in a computational study on the design of real-world gas transport networks.