Finite convergence of DR for PSD boundary with a single-entry hyperplane constraint

Establish whether the Douglas–Rachford algorithm terminates after finitely many iterations for the feasibility problem of finding a real symmetric matrix X in ∂S^n_+ ∩ S_{11}, where ∂S^n_+ = {X ∈ S^n_+ : λ_min(X) = 0} is the boundary of the positive semidefinite cone and S_{11} = {X ∈ S^n : X_{11} = 1} is the hyperplane fixing the (1,1)-entry; if finite convergence occurs, derive a rigorous theoretical explanation for this phenomenon.

Background

Setting 3 in Section 4.2 examines the intersection of the boundary of the PSD cone ∂S^n_+ with the hyperplane S_{11} enforcing X_{11} = 1. The projection onto S_{11} is provided explicitly, and the experiments compare DR, LT, and PLT in this boundary-constrained setting, which is more analogous to the plane curve cases with affine hyperplanes studied earlier.

Empirical results (Figure~\ref{fig:convergence plot_psdb_s11}) indicate that DR appears to converge in finitely many steps, whereas LT and PLT exhibit quadratic-like behaviour rather than finite termination. The authors explicitly state that they have no theoretical explanation for the finite convergence of DR in this setting, marking it as an open issue worthy of theoretical analysis.