Theoretical guarantee of eigenvalue coalescence under continuation

Establish whether increasing the aggregation parameter through a continuation scheme can theoretically guarantee eigenvalue coalescence at the true optimum of a buckling-constrained topology-optimization problem.

Background

The paper explains that finite aggregation parameters can produce eigenvalues that are close but do not truly coalesce, making modality identification unreliable. For the fixed-fixed one-dimensional compressed-column problem, the authors note that obtaining a very small relative eigenvalue difference requires gradually increasing the aggregation parameter to a very large value, which is computationally expensive. The unresolved issue is whether such a continuation procedure can provide a theoretical guarantee that the eigenvalues actually coalesce at the true optimum, rather than merely becoming numerically close.

References

This makes the analysis computationally very expensive, and we still can’t theoretically guarantee the coalescence.

— Topology optimization with buckling constraints: Adaptive eigenvalue aggregation and modality identification  (2609.19603 - Prabha et al., 17 Sep 2026) in Section 4.2, “A typical example for fake coalescence,” page 21