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On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

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  • M. V. Solodov

    (Instituto de Matemática Pura e Aplicada)

Abstract

We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the function and subgradient values involves solving exactly a certain optimization problem, which can be relatively costly (and as we show, in any case unnecessary). We show that approximation with some finite precision is sufficient in this setting and give an explicit characterization of this precision. Alternatively, our result can be viewed as a stability analysis of standard proximal bundle methods, as it answers the following question: for a given approximation error, what kind of approximate solution can be obtained and how does it depend on the magnitude of the perturbation?

Suggested Citation

  • M. V. Solodov, 2003. "On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 151-165, October.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:1:d:10.1023_b:jota.0000005046.70410.02
    DOI: 10.1023/B:JOTA.0000005046.70410.02
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    References listed on IDEAS

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    1. M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
    2. M. V. Solodov, 1997. "Convergence Analysis of Perturbed Feasible Descent Methods," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 337-353, May.
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    Cited by:

    1. Shuai Liu, 2019. "A simple version of bundle method with linear programming," Computational Optimization and Applications, Springer, vol. 72(2), pages 391-412, March.
    2. Fan-Yun Meng & Li-Ping Pang & Jian Lv & Jin-He Wang, 2017. "An approximate bundle method for solving nonsmooth equilibrium problems," Journal of Global Optimization, Springer, vol. 68(3), pages 537-562, July.
    3. W. Hare & C. Sagastizábal & M. Solodov, 2016. "A proximal bundle method for nonsmooth nonconvex functions with inexact information," Computational Optimization and Applications, Springer, vol. 63(1), pages 1-28, January.
    4. N. Hoseini Monjezi & S. Nobakhtian, 2022. "An inexact multiple proximal bundle algorithm for nonsmooth nonconvex multiobjective optimization problems," Annals of Operations Research, Springer, vol. 311(2), pages 1123-1154, April.
    5. Jian Lv & Li-Ping Pang & Fan-Yun Meng, 2018. "A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information," Journal of Global Optimization, Springer, vol. 70(3), pages 517-549, March.
    6. Alireza Hosseini & S. M. Hosseini, 2013. "A New Steepest Descent Differential Inclusion-Based Method for Solving General Nonsmooth Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 698-720, December.
    7. Grégory Emiel & Claudia Sagastizábal, 2010. "Incremental-like bundle methods with application to energy planning," Computational Optimization and Applications, Springer, vol. 46(2), pages 305-332, June.
    8. Jérôme Malick & Welington Oliveira & Sofia Zaourar, 2017. "Uncontrolled inexact information within bundle methods," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 5-29, March.
    9. Xiaoliang Wang & Liping Pang & Qi Wu & Mingkun Zhang, 2021. "An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization," Mathematics, MDPI, vol. 9(8), pages 1-27, April.

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