Convergence rates for SDP discretizations

Establish convergence rates for the finite-dimensional semidefinite-programming discretizations of the infinite-dimensional Lagrange dual problem, potentially by proving higher-than-$H^1$ regularity for the optimal dual variable and the critical fields associated with the positive-sem definiteness constraint.

Background

The paper proves convergence of the SDP discretizations to the infinite-dimensional dual optimum and proves strong convergence of the discrete optimal dual variables. It does not provide quantitative rates. The authors suggest that classical Galerkin approximation rates may apply, but note that proving such rates requires additional regularity of the dual optimizer and of the critical fields governing the semidefinite constraint.

References

First, although we have proved that SDP discretizations of the dual problem \cref{e:ldp} converge, we have no convergence rates. Our computations suggest that classical approximation rates for Galerkin methods should apply. Proving this, however, requires a higher than $H1$ regularity for the optimal dual variable $\xi$ and for the `critical fields' associated with the constraint of \cref{e:ldp}.

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

A second open challenge is to remove the computational bottlenecks in the solution of high-resolution SDP discretizations of \cref{e:ldp}.

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