Determine whether a variable-metric reformulation simplifies PDFRL convergence analysis

Determine whether the convergence of the primal-dual forward-reflected algorithm with linesearch (PDFRL) can be analyzed through a similar variable-metric reformulation that yields a simpler presentation of the convergence result.

Background

The paper develops PDFRL for solving composite monotone inclusions involving a maximally monotone operator, a linearly composed maximally monotone operator, and a single-valued monotone operator that is only locally Lipschitz continuous. Its convergence proof uses a direct Lyapunov-based analysis, which also supports an ergodic convergence-rate result for the associated consensus-penalised gap function.

For the globally Lipschitz case, the authors note that a related primal-dual twice-reflected method admits a representation as a forward-reflected-backward splitting method under a different metric. This observation suggests a possible analogous variable-metric treatment of PDFRL with linesearch, but the paper leaves unresolved whether such a reformulation would actually simplify the presentation of the convergence analysis.