Adding variables while preserving real-zero structure

Construct principled multivariate liftings with additional variables that preserve real-zero-ness, and determine how different variable-expansion mechanisms affect the accuracy of the associated spectrahedral relaxations.

Background

The paper shows that a particular multivariate refinement improves the certified bounds obtained from the spectrahedral relaxation. It also notes that multivariate liftings can be generated through different recursions and stability-preserving operators.

A central unresolved difficulty is to enlarge the number of variables beyond the demonstrated construction without losing real-zero-ness or related properties such as stability and hyperbolicity. Resolving this would enable systematic comparison of competing liftings and their approximation quality.

References

The problem is that it is not entirely clear how to add more variables keeping RZ-ness beyond the explained process of recursion through stability preserving operators.

— Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets  (2507.18434 - Nevado, 24 Jul 2025) in Paragraph following Remark “More variables,” Section 11, “Other venues for improvements”