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An accelerated proximal gradient method for multiobjective optimization

Author

Listed:
  • Hiroki Tanabe

    (Yahoo Japan Corporation)

  • Ellen H. Fukuda

    (Kyoto University)

  • Nobuo Yamashita

    (Kyoto University)

Abstract

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ( $$O(1 / k^2)$$ O ( 1 / k 2 ) ), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

Suggested Citation

  • Hiroki Tanabe & Ellen H. Fukuda & Nobuo Yamashita, 2023. "An accelerated proximal gradient method for multiobjective optimization," Computational Optimization and Applications, Springer, vol. 86(2), pages 421-455, November.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:2:d:10.1007_s10589-023-00497-w
    DOI: 10.1007/s10589-023-00497-w
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    References listed on IDEAS

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    1. M. L. N. Gonçalves & F. S. Lima & L. F. Prudente, 2022. "Globally convergent Newton-type methods for multiobjective optimization," Computational Optimization and Applications, Springer, vol. 83(2), pages 403-434, November.
    2. Hiroki Tanabe & Ellen H. Fukuda & Nobuo Yamashita, 2019. "Proximal gradient methods for multiobjective optimization and their applications," Computational Optimization and Applications, Springer, vol. 72(2), pages 339-361, March.
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    Cited by:

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    2. Douglas S. Gonçalves & Max L. N. Gonçalves & Jefferson G. Melo, 2024. "An away-step Frank–Wolfe algorithm for constrained multiobjective optimization," Computational Optimization and Applications, Springer, vol. 88(3), pages 759-781, July.
    3. Xiaopeng Zhao & Ravi Raushan & Debdas Ghosh & Jen-Chih Yao & Min Qi, 2025. "Proximal gradient method for convex multiobjective optimization problems without Lipschitz continuous gradients," Computational Optimization and Applications, Springer, vol. 91(1), pages 27-66, May.
    4. Qing-Rui He & Chun-Rong Chen & Sheng-Jie Li, 2023. "Spectral conjugate gradient methods for vector optimization problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 457-489, November.
    5. Douglas S. Gonçalves & Max L. N. Gonçalves & Jefferson G. Melo, 2025. "Improved Convergence Rates for the Multiobjective Frank–Wolfe Method," Journal of Optimization Theory and Applications, Springer, vol. 205(2), pages 1-25, May.
    6. Qing-Rui He & Sheng-Jie Li & Bo-Ya Zhang & Chun-Rong Chen, 2024. "A family of conjugate gradient methods with guaranteed positiveness and descent for vector optimization," Computational Optimization and Applications, Springer, vol. 89(3), pages 805-842, December.
    7. Anteneh Getachew Gebrie & Ellen Hidemi Fukuda, 2025. "Adaptive Generalized Conditional Gradient Method for Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-27, July.

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