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Convergence Analysis of Difference-of-Convex Algorithm with Subanalytic Data

Author

Listed:
  • Hoai An Le Thi

    (University of Lorraine)

  • Van Ngai Huynh

    (University of Quynhon)

  • Tao Pham Dinh

    (University of Normandie)

Abstract

Difference-of-Convex programming and related algorithms, which constitute the backbone of nonconvex programming and global optimization, were introduced in 1985 by Pham Dinh Tao and have been extensively developed by Le Thi Hoai An and Pham Dinh Tao since 1994 to become now classic and increasingly popular. That algorithm is a descent method without linesearch and every limit point of its generated sequence is a critical point of the related Difference-of-Convex program. Determining its convergence rate is a challenging problem. Its knowledge is crucial from both theoretical and practical points of view. In this work, we treat this problem for the class of Difference-of-Convex programs with subanalytic data by using the nonsmooth form of the Lojasiewicz inequality. We have successfully proved that the whole sequence is convergent, if it is bounded, provided that the objective function is subanalytic continuous on its domain and one of the two Difference-of-Convex components is differentiable with locally Lipschitz derivative. We also established a result on the convergence rate, which depended on the Lojasiewicz exponent of the objective function. Finally, for both classes of trust-region subproblems and nonconvex quadratic programs, we showed that the Lojasiewicz exponent was one half, and thereby, our proposed algorithms applied to these Difference-of-Convex programs were Root-linearly convergent.

Suggested Citation

  • Hoai An Le Thi & Van Ngai Huynh & Tao Pham Dinh, 2018. "Convergence Analysis of Difference-of-Convex Algorithm with Subanalytic Data," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 103-126, October.
    • Handle: RePEc:spr:joptap:v:179:y:2018:i:1:d:10.1007_s10957-018-1345-y
    DOI: 10.1007/s10957-018-1345-y
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    References listed on IDEAS

    as
    1. Hoai Le Thi & Tao Pham Dinh & Nguyen Yen, 2012. "Behavior of DCA sequences for solving the trust-region subproblem," Journal of Global Optimization, Springer, vol. 53(2), pages 317-329, June.
    2. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    3. Hoang Ngoc Tuan, 2012. "Convergence Rate of the Pham Dinh–Le Thi Algorithm for the Trust-Region Subproblem," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 904-915, September.
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    Cited by:

    1. Min Tao & Jiang-Ning Li, 2023. "Error Bound and Isocost Imply Linear Convergence of DCA-Based Algorithms to D-Stationarity," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 205-232, April.
    2. Hoai An Thi & Thi Minh Tam Nguyen & Tao Pham Dinh, 2023. "On solving difference of convex functions programs with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 86(1), pages 163-197, September.
    3. Tao Pham Dinh & Van Ngai Huynh & Hoai An Le Thi & Vinh Thanh Ho, 2022. "Alternating DC algorithm for partial DC programming problems," Journal of Global Optimization, Springer, vol. 82(4), pages 897-928, April.
    4. Amar Andjouh & Mohand Ouamer Bibi, 2022. "Adaptive Global Algorithm for Solving Box-Constrained Non-convex Quadratic Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 360-378, January.
    5. M. V. Dolgopolik, 2022. "DC Semidefinite programming and cone constrained DC optimization I: theory," Computational Optimization and Applications, Springer, vol. 82(3), pages 649-671, July.
    6. Hongbo Dong & Min Tao, 2021. "On the Linear Convergence to Weak/Standard d-Stationary Points of DCA-Based Algorithms for Structured Nonsmooth DC Programming," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 190-220, April.
    7. M. V. Dolgopolik, 2023. "DC semidefinite programming and cone constrained DC optimization II: local search methods," Computational Optimization and Applications, Springer, vol. 85(3), pages 993-1031, July.

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