Obtain an optimal solution of the general binary optimization model

Obtain an optimal solution of the general binary optimization problem for constructing combinatorial multivector fields from finite vector-field data, specifically the optimization problem in equation (10) (denoted \(\ref{eq:MinOptGenCDS}\)).

Background

The generalized CDS model uses binary variables and convexity constraints to construct combinatorial multivector fields from data. The paper notes that the model is NP-hard and may require post-processing to ensure that feasible solutions induce valid combinatorial multivector fields.

The authors explicitly report that their computational attempt did not reach an optimal solution even after seven days. Thus, obtaining an optimal solution for this concrete optimization problem is left unresolved in the paper, particularly in settings where the model is intended to represent more complex dynamical systems.

References

Finally, one more hurdle is to solve an optimization problem with binary variables is Np-hard. We tried to solve the problem in the subsection \ref{ss:OverviewMain}. After seven days of solving (\ref{eq:MinOptGenCDS}), we still do not have an optimal solution for this problem.

— From Data to Combinatorial Multivector field Through an Optimization-Based Framework  (2501.02023 - Côté et al., 2 Jan 2025) in Section 3, subsection “Model 1: Generalization of the CDS Model”