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On a Unified and Simplified Proof for the Ergodic Convergence Rates of PPM, PDHG and ADMM

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arxiv 2305.02165 v2 pith:IABDIZTY submitted 2023-05-03 math.OC

classification math.OC
keywords algorithmspdhgproofadmmergodicgradientmethodprimal-dual
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abstract

We present a unified viewpoint of proximal point method (PPM), primal-dual hybrid gradient (PDHG) and alternating direction method of multipliers (ADMM) for solving convex-concave primal-dual problems. This viewpoint shows the equivalence of these three algorithms upto a norm change, and it leads to a four-line simple proof of their $\mathcal O(1/k)$ ergodic rates. The simple proof technique is not limited to these three algorithms, but can also be utilized to analyze related algorithms, such as gradient descent, linearized PDHG, inexact algorithms, just to name a few.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the Relationship Between CoCoA and ADMM for Distributed Empirical Risk Minimization

    math.OC 2025-02 conditional novelty 7.0 of 10

    Ridge-regularized CoCoA is shown to be a special case of proximal ADMM, and consensus ADMM is shown to be equivalent to proximal ADMM under a parameter mapping and sign reversal of the saddle objective.

  2. An Overview of GPU-based First-Order Methods for Linear Programming and Extensions

    math.OC 2025-06 unverdicted novelty 2.0 of 10

    A survey of GPU-based first-order LP solvers focusing on cuPDLP, its PDHG core, theory, benchmarks, and extensions to QP, SDP, and conic programming.

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