Convergence of lower-bound relaxations

Determine whether lower bounds on the Lagrange-dual value obtained using null-Lagrangian translations or dual occupation-measure relaxations converge to the desired bound for the optimal cooling problem.

Background

The paper introduces finite-dimensional semidefinite-programming discretizations of the infinite-dimensional Lagrange dual problem. These discretizations converge under density assumptions, but the authors distinguish them from alternative relaxations based on null Lagrangians or occupation measures, for which convergence has not been established. Determining whether those alternative lower-bound constructions converge remains unresolved.

References

Lower bounds on $L(Pe)$ could be computed using null Lagrangian translations or dual `occupation measure' relaxations, but we do not presently know if these lower bounds converge.

— Optimizing bounds for energy-constrained optimal cooling problems in two dimensions  (2608.14334 - Braga et al., 14 Aug 2026) in Section 3, subsection “A general discretization scheme”

While non-sharp bounds may still exhibit the same large-$Pe$ asymptotics as the true optimal cost $J(Pe)$, we do not know any nontrivial case in which this happens.

— Optimizing bounds for energy-constrained optimal cooling problems in two dimensions  (2608.14334 - Braga et al., 14 Aug 2026) in Section 7, Conclusion

However, we cannot determine the asymptotic decay rate of $L(Pe)$ with any confidence because the range of P\ numbers spanned by our computations is too small.

— Optimizing bounds for energy-constrained optimal cooling problems in two dimensions  (2608.14334 - Braga et al., 14 Aug 2026) in Section 5, subsection “Optimal cooling in a uniformly heated annulus”