| Author: | Zhang, Guojun |
| Title: | Fast algorithms for optimal transport problems : theory, acceleration, and extensions |
| Advisors: | Sun, Defeng (AMA) |
| Degree: | Ph.D. |
| Year: | 2026 |
| Department: | Department of Applied Mathematics |
| Pages: | xx, 166 pages : color illustrations |
| Language: | English |
| Abstract: | Optimal Transport (OT) has become a powerful tool for comparing probability distributions, but solving OT-related problems remains computationally intensive. In the discrete setting, many important OT tasks—such as computing the Kantorovich-Wasserstein (KW) distance and Wasserstein barycenters with pre-specified finite supports—can be formulated as large-scale linear programs, which form a special class of linearly constrained convex optimization problems (COPs). This thesis develops scalable, accelerated algorithms based on the Alternating Direction Method of Multipliers (ADMM) to efficiently solve COPs, with applications to these key OT problems as well as extensions to general linear programming (LP). First, we propose a Halpern-Peaceman-Rachford (HPR) algorithm without proximal terms for solving COPs by applying Halpern iteration to the Peaceman-Rachford splitting method. The HPR algorithm achieves a non-ergodic iteration complexity of O(1/ε) for obtaining an ε-approximate solution in terms of the Karush-Kuhn-Tucker (KKT) residual and the objective error. When applied to the Wasserstein barycenter problem with pre-specified finite supports—that is, computing the barycenter of T discrete probability distributions, each supported on m points—we derive closed-form solutions for the subproblems, leading to an overall complexity of O(Tm²/ε) flops. This improves upon the previous best-known bound of O(Tm²/ε) by eliminating extraneous logarithmic factors. Second, we develop an accelerated preconditioned ADMM (pADMM) method with semi-proximal terms for solving COPs, where the HPR algorithm arises as a special case when the proximal terms are removed. To accelerate pADMM, we reformulate it as a proximal point algorithm (PPA) equipped with a positive semidefinite, possibly degenerate, preconditioner, due to the lack of strong convexity of the proximal terms in the pADMM. By integrating Halpern iteration and fast Krasnosel'ski˘ı-Mann iteration into the resulting degenerate PPA (dPPA), we obtain asymptotic o(1/k) and non-asymptotic O(1/k) iteration complexity. Building on this, we develop an accelerated pADMM algorithm that achieves these rates for the KKT residual and the objective error. We further apply this framework to develop HOT, a Halpern accelerating algorithm for computing the KW distance. Instead of solving the original OT problem directly, HOT solves an equivalent reduced formulation that leverages the problem structure to reduce computational complexity. For the discrete OT problem with m support points in R², under an L₂²-ground cost, we derive closed-form solutions for HOT's subproblems, resulting in an overall computational complexity of O(m1.5/ε) flops—surpassing the previous best-known bound of O(m²/ε). Finally, inspired by the promising performance of the HPR method for solving OT-related problems, we introduce HPR-LP, an implementation of the HPR method with semi-proximal terms for solving large-scale general LP problems. Based on the O(1/k) complexity results of the HPR method, we design an adaptive strategy of restart and penalty parameter update to improve the efficiency and robustness of the HPR method. We conduct extensive numerical experiments on different LP benchmark datasets using an NVIDIA A100-SXM4-80GB GPU under different stopping tolerances. The Julia implementation of HPR-LP achieves a 2.39x to 5.70x speedup measured by SGM10 on benchmark datasets with presolve (2.03x to 4.06x without presolve) over the award-winning solver PDLP with the tolerance of 10⁻⁸. |
| Rights: | All rights reserved |
| Access: | open access |
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