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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Accelerated Uzawa methods for convex optimization
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by Min Tao and Xiaoming Yuan PDF
Math. Comp. 86 (2017), 1821-1845 Request permission

Abstract:

We focus on a linearly constrained strongly convex minimization model and discuss the application of the classical Uzawa method. Our principal goal is to show that some existing acceleration schemes can be used to accelerate the Uzawa method in the sense that the worst-case convergence rate (measured by the iteration complexity) of the resulting accelerated Uzawa schemes is $O(1/k^2)$ where $k$ represents the iteration counter. Our discussion assumes that the objective function is given by a black-box oracle; thus an inexact version of the Uzawa method with a sequence of dynamically-chosen step sizes is implemented. A worst-case convergence rate of $O(1/k)$ is also shown for this inexact version. Some preliminary numerical results are reported to verify the acceleration effectiveness of the accelerated Uzawa schemes and their superiority over some existing methods.
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Additional Information
  • Min Tao
  • Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
  • MR Author ID: 987506
  • Email: taom@nju.edu.cn
  • Xiaoming Yuan
  • Affiliation: Department of Mathematics, Hong Kong Baptist University, Hong Kong, People’s Republic of China
  • MR Author ID: 729439
  • Email: xmyuan@hkbu.edu.hk
  • Received by editor(s): March 31, 2015
  • Received by editor(s) in revised form: January 28, 2016
  • Published electronically: October 20, 2016
  • Additional Notes: The first author was supported by the Natural Science Foundation of China grant NSFC-11301280, and the Fundamental Research Funds for the Central Universities, 020314330019
    The second author was supported in part by the General Research Fund from Hong Kong Research Grants Council, HKBU 12300515
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1821-1845
  • MSC (2010): Primary 90C25; Secondary 94A08
  • DOI: https://doi.org/10.1090/mcom/3145
  • MathSciNet review: 3626538