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On the minimization of possibly discontinuous functions by means of pointwise approximations

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Abstract

A general approach for the solution of possibly discontinuous optimization problems by means of pointwise (perhaps smooth) approximations will be proposed. It will be proved that sequences generated by pointwise approximation techniques eventually satisfy well justified stopping criteria. Numerical examples will be given.

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References

  1. Abraham, Z., Xin, F., Tan, P.-N.: Smoothed quantile regression for statistical downscaling of extreme events in climate modeling. In: Srivastava, A.N., Chawla, N.V., Perera, A.S. (eds.) Proceedings of the 2011 Conference on Intelligent Data Understanding (CIDU 2011), pp. 92–106. Mountain View, CA (2011)

  2. Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On Augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60, 627–641 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Birgin, E.G., Castelani, E.V., Martinez, A.L.M., Martínez, J.M.: Outer trust-region method for constrained optimization. J. Optim. Theory Appl. 150, 142–155 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Birgin, E.G., Floudas, C.A., Martínez, J.M.: Global minimization using an Augmented Lagrangian method with variable lower-level constraints. Math. Program. 125, 139–162 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Birgin, E.G., Martínez, J.M.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 22, 101–125 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. SIAM, Philadelphia (2014)

    Book  MATH  Google Scholar 

  8. Castelani, E.V., Martinez, A.L., Martínez, J.M., Svaiter, B.F.: Addressing the greediness phenomenon in nonlinear programming by means of proximal Augmented Lagrangians. Comput. Optim. Appl. 46, 229–245 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Heuts, R., de Klein, J.: An (s-q) inventory model with stochastic and interrelated lead times. Naval Res. Logist. 42, 839–859 (1995)

    Article  MATH  Google Scholar 

  11. Kamimura, K., Nishino, H.: An efficient method for determining economical configurations of elementary packet-switched networks. IEEE Trans. Cybern. 39, 278–288 (1991)

    Google Scholar 

  12. Maia, L.O.A., Vidal de Carvalho, L.A., Qassim, R.Y.: Synthesis of utility systems by simulated annealing. Comput. Chem. Eng. 19, 481–488 (1995)

    Article  Google Scholar 

  13. Martínez, J.M.: Minimization of discontinuous cost functions by smoothing. Acta Appl. Math. 71, 245–260 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rozvany, G.: Optimal plastic design with discontinuous cost functions. J. Appl. Mech. 41, 309–310 (1974)

    Article  Google Scholar 

  16. Shulman, A., Vachani, R.: A decomposition algorithm for capacity expansion of local access networks. IEEE Trans. Cybern. 41, 1063–1073 (1993)

    MATH  Google Scholar 

  17. Turkay, M., Grossmann, I.E.: Disjunctive programming techniques for the optimization of process systems with discontinuous investment costs-multiple size regions. Ind. Eng. Chem. Res. 35, 2611–2623 (1996)

    Article  Google Scholar 

  18. Vicente, L.N., Custódio, A.L.: Analysis of direct searches for discontinuous functions. Math. Program. 133, 299–325 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wechsung, A., Barton, P.I.: Global optimization of bounded factorable functions with discontinuities. J. Global Optim. 58, 1–30 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zang, I.: Discontinuous optimization by smoothing. Math. Oper. Res. 6, 140–152 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the associate editor and the three referees for their comments and remarks that helped a lot to improve the first version of this work.

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Authors and Affiliations

  1. Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP, 05508-090, Brazil

    E. G. Birgin

  2. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Serbia

    N. Krejić

  3. Department of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific Computing, State University of Campinas, Campinas, SP, 13083-859, Brazil

    J. M. Martínez

Authors
  1. E. G. Birgin
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  2. N. Krejić
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  3. J. M. Martínez
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Corresponding author

Correspondence to E. G. Birgin.

Additional information

This work has been partially supported by the Brazilian agencies FAPESP (Grants 2010/10133-0, 2013/03447-6, 2013/05475-7, 2013/07375-0, and 2014/18711-3) and CNPq (Grants 309517/2014-1 and 303750/2014-6) and by the Serbian Ministry of Education, Science, and Technological Development (Grant 174030).

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Birgin, E.G., Krejić, N. & Martínez, J.M. On the minimization of possibly discontinuous functions by means of pointwise approximations. Optim Lett 11, 1623–1637 (2017). https://doi.org/10.1007/s11590-016-1068-7

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  • DOI: https://doi.org/10.1007/s11590-016-1068-7

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