Finite convergence of DR and LT for PSD cone with unit-diagonal constraint

Establish whether the Douglas–Rachford algorithm and the Lyapunov surrogate method terminate after finitely many iterations for the feasibility problem of finding a real symmetric matrix X in S^n_+ ∩ S_1, where S^n_+ = {X ∈ S^n : v^T X v ≥ 0 for all v ∈ ℝ^n} is the positive semidefinite cone and S_1 = {X ∈ S^n : diag(X) = 1} is the affine subspace of unit-diagonal matrices; if such finite convergence occurs, derive a rigorous theoretical explanation for this phenomenon.

Background

In Section 4.2, the paper investigates semidefinite feasibility problems as higher-dimensional analogues of plane curve intersections. Setting 1 considers the intersection of the positive semidefinite cone S^n_+ with the affine subspace S_1 consisting of symmetric matrices whose diagonal entries equal 1. The Douglas–Rachford operator is taken as T_{B,A} := ½(I + R_A R_B) with A = S^n_+ and B = S_1, and projections onto both sets are provided.

Numerical experiments reported in Figure~\ref{fig:convergence plot_psd_s1} show that, in this setting, both the Douglas–Rachford method (DR) and the Lyapunov surrogate method (LT) appear to terminate after finitely many steps, while the projected variant (PLT) does not terminate finitely and instead displays behaviour consistent with quadratic convergence. Despite these observations, the authors explicitly state that they lack a theoretical explanation for the finite termination of DR and LT in this setting, motivating a precise theoretical investigation.