Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
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ON AUGMENTED LAGRANGIAN METHODS OF MULTIPLIERS AND ALTERNATING DIRECTION METHODS OF MULTIPLIERS FOR MATRIX OPTIMIZATION PROBLEMS

  • Gue Myung, Lee (Department of Applied Mathematics, Pukyong National University) ;
  • Jae Hyoung, Lee (Department of Applied Mathematics, Pukyong National University)
  • Received : 2022.04.06
  • Accepted : 2022.07.20
  • Published : 2022.12.06
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Abstract

In this paper, we consider matrix optimization problems. We investigate augmented Lagrangian methods of multipliers and alternating direction methods of multipliers for the problems. Following the proofs of Eckstein [3], and Eckstein and Yao [5], we prove convergence theorems for augmented Lagrangian methods of multipliers and alternating direction methods of multipliers for the problems.

Keywords

Acknowledgement

This research was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (NRF-2017R1E1A1A03069931).

References

  1. A. Beck, Quadratic Matrix Programming, SIAM J. Optim., 17 (2006), 1224-1238. https://doi.org/10.1137/05064816X
  2. S, Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, Found. Trends Machine Learning, 3(1) (2010), 1-122. https://doi.org/10.1561/2200000016
  3. J. Eckstein, Augmented Lagrangian and Alternating Direction Methods for Convex Optimization: A Tutorial and Some Illustrative Computational Results, RRR 32-2012, December 2012.
  4. J. Eckstein and D.P. Bertsekas, On the Douglas-Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators, Math. Prog., 55 (1992), 293-318. https://doi.org/10.1007/BF01581204
  5. J. Eckstein and W. Yao, Understanding the Convergence of the Alternating Direction Method of Multipliers: Theoretical and Computational Perspectives, Pac. J. Optim., 11(4) (2015), 619-644.
  6. V. Jeyakumar, G.M. Lee and N. Dinh, New Sequential Lagrange Multiplier Conditions Characterizing Optimality without Constraint Qualification for Convex Programs, SIAM J. Optim., 14 (2003), 534-547. https://doi.org/10.1137/S1052623402417699
  7. M.A. Krasnosel'skii, Two Remarks on the Method of Successive Approximations, Uspekhi Mat. Nauk, 10(1) (1996), 123-127.
  8. W.R. Mann, Mean Value Methods in Iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  9. L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Rev., 38 (1996), 49-95. https://doi.org/10.1137/1038003