On the ergodic convergence properties of the Peaceman-Rachford method and their applications in solving linear programming
Keywords: Peaceman-Rachford method, ergodic convergence properties, linear programming, compleixty, parallel computing
Abstract: In this paper, we study the ergodic convergence properties of the Peaceman-Rachford (PR) method with semi-proximal terms for solving convex optimization problems (COPs). By reformulating the PR method as a degenerate proximal point method, for the first time we establish the global convergence of the ergodic sequence generated by the PR method with broadly chosen semi-proximal terms under the assumption that there exists a Karush–Kuhn–Tucker (KKT) solution to the COPs. This result represents a significant departure from previous studies on the non-ergodic convergence of the PR method, which typically requires strong convexity (or strong monotonicity in the reformulated operator) conditions that are hardly satisfied for COPs. Moreover, we establish an ergodic iteration complexity of $O(1/k)$ of the PR method with semi-proximal terms, measured by the objective error, the feasibility violation, and the KKT residual using the $\varepsilon$-subdifferential. Based on these convergence properties, we introduce the solver EPR-LP, using the ergodic sequence of the PR method with semi-proximal terms for solving linear programming (LP) problems. EPR-LP incorporates an adaptive restart strategy and dynamic penalty parameter updates for efficiency and robustness. Extensive numerical experiments on LP benchmark datasets, executed on a high-performance GPU, show that our Julia-based solver outperforms the award-winning solver PDLP at a tolerance level of $10^{-8}$.
Primary Area: optimization
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Submission Number: 11165
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