In this paper, we aim to accelerate a preconditioned alternating direction method of multipliers (pADMM), whose proximal terms are convex quadratic functions, for solving linearly constrained convex optimization problems. To achieve this, we first reformulate the pADMM into a form of the proximal point method (PPM) with a positive semidefinite preconditioner which can be degenerate due to the lack of strong convexity of the proximal terms in the pADMM. Then we accelerate the pADMM by accelerating the reformulated degenerate PPM (dPPM). Specifically, we first propose an accelerated dPPM by integrating the Halpern iteration and the fast Krasnosel’skiĭ–Mann iteration into it, achieving asymptotic [Formula: see text] and nonasymptotic [Formula: see text] convergence rates. Subsequently, building upon the accelerated dPPM, we develop an accelerated pADMM algorithm that exhibits both asymptotic [Formula: see text] and nonasymptotic [Formula: see text] nonergodic convergence rates concerning the Karush–Kuhn–Tucker residual and the primal objective function value gap. Preliminary numerical experiments validate the theoretical findings, demonstrating that the accelerated pADMM outperforms the pADMM in solving convex quadratic programming problems.

This paper proposes an efficient HOT algorithm for solving the optimal transport (OT) problems with finite supports. We particularly focus on an efficient implementation of the HOT algorithm for the case where the supports are in $\mathbb {R}^{2}$ with ground distances calculated by $L_{2}^{2}$-norm. Specifically, we design a Halpern accelerating algorithm to solve the equivalent reduced model of the discrete OT problem. Moreover, we derive a novel procedure to solve the involved linear systems in the HOT algorithm in linear time complexity. Consequently, we can obtain an $\varepsilon$-approximate solution to the optimal transport problem with $M$ supports in $O(M^{1.5}/\varepsilon )$ flops, which significantly improves the best-known computational complexity. We further propose an efficient procedure to recover an optimal transport plan for the original OT problem based on a solution to the reduced model, thereby overcoming the limitations of the reduced OT model in applications that require the transport plan. We implement the HOT algorithm in PyTorch and extensive numerical results show the superior performance of the HOT algorithm compared to existing state-of-the-art algorithms for solving the OT problems.

With numerous renewable generators and energy storage systems integrated into the power grids, the security-constrained DC optimal power flow (DCOPF) is essential for power system operation. For large-scale power grids, traditional CPU-based optimization algorithms (such as the simplex and barrier methods) have saturated in computational efficiency and are inherently difficult to parallelize. To tackle these issues, by incorporating the symmetric Gauss–Seidel (sGS) decomposition, this work develops a GPU-based Halpern Peaceman-Rachford algorithm, termed the sGS-HPR, which enjoys an $O(1/k)$ iteration complexity in terms of the KKT residual. Moreover, the closed-form solutions for all subproblems are derived, which only consist of matrix-vector multiplications and vector operations, and thus can be easily parallelized on GPUs. As a consequence, the developed sGS-HPR algorithm enjoys a $O({N_{L} \times n} / \epsilon)$ non-ergodic computational complexity in terms of floating-point operations for obtaining an $\epsilon$-optimal solution measured by the KKT residual for large-scale DCOPF problems, where $n$ represents the variable dimension, and $N_{L}$ denotes the number of branches in the power grid. Extensive numerical tests on large-scale power grids, reaching up to the 9241-bus PEGASE system, demonstrate the scalability and superior efficiency of the developed GPU-based sGS-HPR algorithm compared to state-of-the-art methods. Notably, the proposed method achieves a 6× speedup compared with Gurobi for large-scale instances. Additionally, for ultra-large-scale cases, Gurobi throws an “out-of-memory” error, while the proposed sGS-HPR algorithm maintains its computational scalability and efficiency.