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An Improvement of the Alternating Direction Method of Multipliers to Solve the Convex Optimization Problem

Author

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  • Jingjing Peng

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

  • Zhijie Wang

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

  • Siting Yu

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

  • Zengao Tang

    (College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China)

Abstract

The alternating direction method is one of the attractive approaches for solving convex optimization problems with linear constraints and separable objective functions. Experience with applications has shown that the number of iterations depends significantly on the penalty parameter for the linear constraint. The penalty parameters in the classical alternating direction method are a constant. In this paper, an improved alternating direction method is proposed, which not only adaptively adjusts the penalty parameters per iteration based on the iteration message but also adds relaxation factors to the Lagrange multiplier update steps. Preliminary numerical experiments show that the technique of adaptive adjusting of the penalty parameters per iteration and attaching relaxation factors in the Lagrange multiplier updating steps are effective in practical applications.

Suggested Citation

  • Jingjing Peng & Zhijie Wang & Siting Yu & Zengao Tang, 2025. "An Improvement of the Alternating Direction Method of Multipliers to Solve the Convex Optimization Problem," Mathematics, MDPI, vol. 13(5), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:811-:d:1602587
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    References listed on IDEAS

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    1. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    2. Deren Han & Defeng Sun & Liwei Zhang, 2018. "Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 622-637, May.
    3. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
    4. Bin Gao & Feng Ma, 2018. "Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 178-204, January.
    5. Jianchao Bai & Jicheng Li & Fengmin Xu & Hongchao Zhang, 2018. "Generalized symmetric ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 70(1), pages 129-170, May.
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