The degenerate variable metric proximal point algorithm and adaptive stepsizes for primal-dual Douglas-Rachford

• Published in: Optimization Vol. 74; no. 6; pp. 1355 - 1381
• Main Authors: Lorenz, Dirk A., Marquardt, Jannis, Naldi, Emanuele
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 26.04.2025; Taylor & Francis LLC
• Subjects: Adaptive algorithms; adaptive stepsizes; Algorithms; Convergence; Douglas-Rachford method; non-stationary primal-dual method; Preconditioned proximal point algorithm; varying preconditioners
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2024.2325552

In this paper, the degenerate preconditioned proximal point algorithm will be combined with the idea of varying preconditioners leading to the degenerate variable metric proximal point algorithm. The weak convergence of the resulting iteration will be proven. From the perspective of the degenerate variable metric proximal point algorithm, a version of the primal-dual Douglas-Rachford method with varying preconditioners will be derived and a proof of its weak convergence which is based on the previous results for the proximal point algorithm, is provided, too. After that, we derive a heuristic on how to choose those varying preconditioners in order to increase the convergence speed of the method.