Theoretical justification of the proximal-parameter effect on convergence speed

Establish a theoretical result guaranteeing the empirical observation that the convergence speed of the Inertial Douglas-Rachford DC algorithm increases as the proximal parameter \(\beta\) decreases.

Background

The numerical experiments examine the Inertial Douglas-Rachford DC algorithm for different choices of the proximal parameter $\beta$. The reported plots suggest that smaller values of $\beta$ lead to larger residual values $\|A\cdot-b\|$ at a given iteration count and, according to the authors, indicate an increase in convergence speed as $\beta$ decreases.

The paper does not provide a theoretical convergence-rate result that explains or guarantees this parameter-dependent behavior. Establishing such a result would connect the observed numerical phenomenon with the convergence theory of the proposed algorithm and would support principled parameter selection in future implementations.