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New global optimality conditions for nonsmooth DC optimization problems

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  • M. V. Dolgopolik

    (Russian Academy of Sciences)

Abstract

In this article we propose a new approach to an analysis of DC optimization problems. This approach was largely inspired by codifferential calculus and the method of codifferential descent and is based on the use of a so-called affine support set of a convex function instead of the Frenchel conjugate function. With the use of affine support sets we define a global codifferential mapping of a DC function and derive new necessary and sufficient global optimality conditions for DC optimization problems. We also provide new simple necessary and sufficient conditions for the global exactness of the $$\ell _1$$ℓ1 penalty function for DC optimization problems with equality and inequality constraints and present a series of simple examples demonstrating a constructive nature of the new global optimality conditions. These examples show that when the optimality conditions are not satisfied, they can be easily utilised in order to find “global descent” directions of both constrained and unconstrained problems. As an interesting theoretical example, we apply our approach to the analysis of a nonsmooth problem of Bolza.

Suggested Citation

  • M. V. Dolgopolik, 2020. "New global optimality conditions for nonsmooth DC optimization problems," Journal of Global Optimization, Springer, vol. 76(1), pages 25-55, January.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00833-7
    DOI: 10.1007/s10898-019-00833-7
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    References listed on IDEAS

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    1. M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 728-744, March.
    2. A. Bagirov & J. Ugon, 2011. "Codifferential method for minimizing nonsmooth DC functions," Journal of Global Optimization, Springer, vol. 50(1), pages 3-22, May.
    3. Hoai Le Thi & Tao Pham Dinh & Huynh Ngai, 2012. "Exact penalty and error bounds in DC programming," Journal of Global Optimization, Springer, vol. 52(3), pages 509-535, March.
    4. Strekalovsky, Alexander S., 2015. "On local search in d.c. optimization problems," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 73-83.
    5. H. Tuy, 2003. "On Global Optimality Conditions and Cutting Plane Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 201-216, July.
    6. Qinghua Zhang, 2013. "A new necessary and sufficient global optimality condition for canonical DC problems," Journal of Global Optimization, Springer, vol. 55(3), pages 559-577, March.
    7. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico & Adil M. Bagirov, 2018. "Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations," Journal of Global Optimization, Springer, vol. 71(1), pages 37-55, May.
    8. M. V. Dolgopolik, 2018. "A convergence analysis of the method of codifferential descent," Computational Optimization and Applications, Springer, vol. 71(3), pages 879-913, December.
    9. Giancarlo Bigi & Antonio Frangioni & Qinghua Zhang, 2010. "Outer approximation algorithms for canonical DC problems," Journal of Global Optimization, Springer, vol. 46(2), pages 163-189, February.
    10. M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 745-762, March.
    11. Alexander S. Strekalovsky, 2017. "Global Optimality Conditions in Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 770-792, June.
    12. R. Horst & N. V. Thoai, 1999. "DC Programming: Overview," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 1-43, October.
    13. Kaisa Joki & Adil M. Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2017. "A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes," Journal of Global Optimization, Springer, vol. 68(3), pages 501-535, July.
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    Cited by:

    1. Yingrang Xu & Shengjie Li, 2022. "Optimality and Duality for DC Programming with DC Inequality and DC Equality Constraints," Mathematics, MDPI, vol. 10(4), pages 1-14, February.
    2. Kabgani, Alireza & Soleimani-damaneh, Majid, 2022. "Semi-quasidifferentiability in nonsmooth nonconvex multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 35-45.
    3. Abdelghali Ammar & Mohamed Laghdir & Ahmed Ed-dahdah & Mohamed Hanine, 2023. "Approximate Subdifferential of the Difference of Two Vector Convex Mappings," Mathematics, MDPI, vol. 11(12), pages 1-14, June.

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