충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제37권1호
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  • Pages.41-55
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  • 2024
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  • 1226-3524(pISSN)
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  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
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GENERALIZED PROXIMAL ITERATIVELY REWEIGHTED ℓ1 ALGORITHM WITH CO-COERCIVENESS FOR NONSMOOTH AND NONCONVEX MINIMIZATION PROBLEM

  • Myeongmin Kang (Department of Mathematics Chungnam National University)
  • 투고 : 2024.01.11
  • 심사 : 2024.01.29
  • 발행 : 2024.02.29
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초록

The nonconvex and nonsmooth optimization problem has been widely applicable in image processing and machine learning. In this paper, we propose an extension of the proximal iteratively reweighted ℓ1 algorithm for nonconvex and nonsmooth minmization problem. We assume the co-coerciveness of a term of objective function instead of Lipschitz gradient condition, which is generalized property of Lipschitz continuity. We prove the global convergence of the proposed algorithm. Numerical results show that the proposed algorithm converges faster than original proximal iteratively reweighed algorithm and existing algorithms.

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과제정보

Supported (in part) by NRF, Project No. NRF-2019R1I1A3A01055168.

참고문헌

  1. H. Attouch, J. Bolte, B.F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program., 137(2013), 91-129. https://doi.org/10.1007/s10107-011-0484-9
  2. J. Bolte, and A. Daniilidis, and A. Lewis, The Lojasiewicz inequality for non-smooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17(2007), 1205-1223. https://doi.org/10.1137/050644641
  3. E. J. Candes, M. B. Wakin, and S. P. Boyd, Enhancing sparsity by reweighted ℓ1 minimization, J. Fourier Anal. Appl., 14(2008), 877-905. https://doi.org/10.1007/s00041-008-9045-x
  4. P. Gong, C. Zhang, Z. Lu, J. Huang, and J. Ye, A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems, international conference on machine learning, 2013.
  5. H. Mingyi, Z.Q. Luo, and M. Razaviyayn, Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM J. Optim., 26(2016), 337-364. https://doi.org/10.1137/140990309
  6. A. Kaplan, and R. Tichatschke, Proximal point methods and nonconvex optimization, J. Glob. Optim., 13(1998), 389-406. https://doi.org/10.1023/A:1008321423879
  7. K. Kurdyka, On gradients of functions definable in o-minimal structures, Annales. Institut. Fourier, 48(1998), 769-783. https://doi.org/10.5802/aif.1638
  8. S. Lojasiewicz, Sur la g'eom'etrie semi-et sous-analytique, Annales. Institut. Fourier, 43(1993), 1575-1595. https://doi.org/10.5802/aif.1384
  9. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Springer, 2004.
  10. P. Ochs, A. Dosovitskiy, T. Brox, and T. Pock, On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision, SIAM J. Imaging Sci., 8(2015), 331-372. https://doi.org/10.1137/140971518
  11. L. P. D. Van den Dries, Tame topology and o-minimal structures, Cambridge university press, 1998.
  12. A. Wilkie, Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function, J. Amer. Math. Soc., 9(1996), 1051-1094. https://doi.org/10.1090/S0894-0347-96-00216-0
  13. J. Yeo, M. Kang, Proximal Linearized Iteratively Reweighted Algorithms for Nonconvex and Nonsmooth Optimization Problem, Axioms, 11(2022), 201.
  14. W. Yu, W. Yin, and J. Zeng, Global convergence of ADMM in nonconvex non-smooth optimization, J. Sci. Comput., 78(2019), 29-63. https://doi.org/10.1007/s10915-018-0757-z