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A two-level distributed algorithm for nonconvex constrained optimization

Author

Listed:
  • Kaizhao Sun

    (Georgia Institute of Technology)

  • X. Andy Sun

    (Massachusetts Institute of Technology)

Abstract

This paper aims to develop distributed algorithms for nonconvex optimization problems with complicated constraints associated with a network. The network can be a physical one, such as an electric power network, where the constraints are nonlinear power flow equations, or an abstract one that represents constraint couplings between decision variables of different agents. Despite the recent development of distributed algorithms for nonconvex programs, highly complicated constraints still pose a significant challenge in theory and practice. We first identify some difficulties with the existing algorithms based on the alternating direction method of multipliers (ADMM) for dealing with such problems. We then propose a reformulation that enables us to design a two-level algorithm, which embeds a specially structured three-block ADMM at the inner level in an augmented Lagrangian method framework. Furthermore, we prove the global and local convergence as well as iteration complexity of this new scheme for general nonconvex constrained programs, and show that our analysis can be extended to handle more complicated multi-block inner-level problems. Finally, we demonstrate with computation that the new scheme provides convergent and parallelizable algorithms for various nonconvex applications, and is able to complement the performance of the state-of-the-art distributed algorithms in practice by achieving either faster convergence in optimality gap or in feasibility or both.

Suggested Citation

  • Kaizhao Sun & X. Andy Sun, 2023. "A two-level distributed algorithm for nonconvex constrained optimization," Computational Optimization and Applications, Springer, vol. 84(2), pages 609-649, March.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:2:d:10.1007_s10589-022-00433-4
    DOI: 10.1007/s10589-022-00433-4
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    References listed on IDEAS

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    1. Min Li & Defeng Sun & Kim-Chuan Toh, 2015. "A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-19.
    2. D’Ambrosio, Claudia & Lodi, Andrea & Wiese, Sven & Bragalli, Cristiana, 2015. "Mathematical programming techniques in water network optimization," European Journal of Operational Research, Elsevier, vol. 243(3), pages 774-788.
    3. Caihua Chen & Yuan Shen & Yanfei You, 2013. "On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, October.
    4. Deren Han & Xiaoming Yuan, 2012. "A Note on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 227-238, October.
    5. Zaiwen Wen & Xianhua Peng & Xin Liu & Xiaoling Sun & Xiaodi Bai, 2013. "Asset Allocation under the Basel Accord Risk Measures," Papers 1308.1321, arXiv.org.
    6. Burak Kocuk & Santanu S. Dey & X. Andy Sun, 2016. "Strong SOCP Relaxations for the Optimal Power Flow Problem," Operations Research, INFORMS, vol. 64(6), pages 1177-1196, December.
    7. Bo Jiang & Tianyi Lin & Shiqian Ma & Shuzhong Zhang, 2019. "Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis," Computational Optimization and Applications, Springer, vol. 72(1), pages 115-157, January.
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