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Essentials of numerical nonsmooth optimization

Author

Listed:
  • Manlio Gaudioso

    (DIMES, Università della Calabria)

  • Giovanni Giallombardo

    (DIMES, Università della Calabria)

  • Giovanna Miglionico

    (DIMES, Università della Calabria)

Abstract

Approximately 60 years ago two seminal findings, the cutting plane and the subgradient methods, radically changed the landscape of mathematical programming. They provided, for the first time, the practical chance to optimize real functions of several variables characterized by kinks, namely by discontinuities in their derivatives. Convex functions, for which a superb body of theoretical research was growing in parallel, naturally became the main application field of choice. The aim of the paper is to give a concise survey of the key ideas underlying successive development of the area, which took the name of numerical nonsmooth optimization. The focus will be, in particular, on the research mainstreams generated under the impulse of the two initial discoveries.

Suggested Citation

  • Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    • Handle: RePEc:spr:aqjoor:v:18:y:2020:i:1:d:10.1007_s10288-019-00425-x
    DOI: 10.1007/s10288-019-00425-x
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    References listed on IDEAS

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    1. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2018. "Minimizing Piecewise-Concave Functions Over Polyhedra," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 580-597, May.
    2. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2009. "On solving the Lagrangian dual of integer programs via an incremental approach," Computational Optimization and Applications, Springer, vol. 44(1), pages 117-138, October.
    3. 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.
    4. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2006. "An Incremental Method for Solving Convex Finite Min-Max Problems," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 173-187, February.
    5. G. Di Pillo & L. Grippo & S. Lucidi, 1997. "Smooth Transformation of the Generalized Minimax Problem," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 1-24, October.
    6. 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.
    7. de GHELLINCK, Guy & VIAL, Jean-Philippe, 1986. "A polynomial Newton method for linear programming," LIDAM Reprints CORE 724, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. J. L. Goffin & A. Haurie & J. P. Vial, 1992. "Decomposition and Nondifferentiable Optimization with the Projective Algorithm," Management Science, INFORMS, vol. 38(2), pages 284-302, February.
    9. Gaudioso, Manlio & Giallombardo, Giovanni & Mukhametzhanov, Marat, 2018. "Numerical infinitesimals in a variable metric method for convex nonsmooth optimization," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 312-320.
    10. Welington Oliveira, 2019. "Proximal bundle methods for nonsmooth DC programming," Journal of Global Optimization, Springer, vol. 75(2), pages 523-563, October.
    11. Demyanov, Alexey V. & Fuduli, Antonio & Miglionico, Giovanna, 2007. "A bundle modification strategy for convex minimization," European Journal of Operational Research, Elsevier, vol. 180(1), pages 38-47, July.
    12. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, December.
    13. Nesterov, Y., 1995. "Complexity estimates of some cutting plane methods based on the analytic barrier," LIDAM Reprints CORE 1167, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. A. M. Bagirov & B. Karasözen & M. Sezer, 2008. "Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 317-334, May.
    15. de GHELLI NCK, G. & VIAL, J.-Ph., 1986. "A polynomial Newton method for linear programming," LIDAM Discussion Papers CORE 1986014, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Lemaréchal, C. & Nemirovskii, A. & Nesterov, Y., 1995. "New variants of bundle methods," LIDAM Reprints CORE 1166, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    18. Hoang Tuy, 2016. "Convex Analysis and Global Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-31484-6, September.
    19. Napsu Karmitsa, 2015. "Diagonal Bundle Method for Nonsmooth Sparse Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 889-905, September.
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    Cited by:

    1. Yves Crama & Michel Grabisch & Silvano Martello, 2022. "Preface," Annals of Operations Research, Springer, vol. 314(1), pages 1-3, July.
    2. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    3. Yves Crama & Michel Grabisch & Silvano Martello, 2021. "4OR comes of age," 4OR, Springer, vol. 19(1), pages 1-13, March.

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