Explain the computational intractability of OBBT on the baseline discretization

Determine why the optimization-based bound-tightening (OBBT) procedure generates computationally demanding linear programs and becomes intractable on the baseline discretization, despite the application of the proposed acceleration techniques.

Background

The paper studies McCormick relaxations for nonconvex PDE-constrained optimization and uses OBBT to tighten state-variable bounds. Although the proposed warm-start and grid-ordering strategies enable OBBT to scale to finer meshes, the authors report a contrasting computational difficulty on the baseline discretization used for comparison with earlier work. Thousands of linear programs must be solved, and the generated LPs become substantially more demanding, preventing completion of even one sweep within the 72-hour computational limit.

The unresolved issue is particularly notable because the authors observed the same behavior both before and after transferring their acceleration techniques to the baseline discretization. Identifying its cause is necessary to understand the method’s scalability and to extend the approach beyond the academic-size meshes considered in the paper.

References

In this case, LPs with a substantial computational demand were generated during the OBBT Algorithm, thereby making the overall algorithm execution intractable since thousands of LPs have to be solved and we had to cancel it after 72 hours with zero completed sweeps. We have not figured out why this is the case yet and have observed this before and after transferring our acceleration techniques to the baseline discretization.

— McCormick relaxations for PDE-constrained optimization on multi-dimensional domains  (2609.08614 - Pokotylo et al., 8 Sep 2026) in Section 4, subsection “Qualitative Assessment of the Bounds”