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Smoothing Approximations to Non-smooth Functions

Year 2018, Volume: 1 Issue: 2, 69 - 74, 21.01.2019

Ahmet Sahiner , Nurullah Yilmaz , Shehab Ahmed Ibrahem

Abstract

In this study, a new smoothing method is proposed for non-smooth functions. The theoretical results and error estimates are presented about this new smoothing method. Finally, some numerical examples are given.

Keywords

non-smooth functions smoothing techniques

References

  • R. T. Rockafellar, R. J-B. Wets R. J-B., Variational Analysis, Springer, Berlin, 1998.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
  • V. F. Demyanov, A. M. Rubinov, Constructive Nonsmooth Analysis. Verlag Peter Lang, Frankfurt, 1995.
  • B. Akteke-Ozkurt, G. W. Weber, G. Koksal, Optimization of generalized desirability functions under model uncertainty, Optimization, 66 (12) 2017, 2157--2169.
  • A. Sahiner, N. Yilmaz and G. Kapusuz, A novel modeling and smoothing technique in global optimization, J. Ind. Manage. Optim., 15 (1) 2019, 113-130.
  • D. Bertsekas, Nondifferentiable optimization via approximation, Math. Program. Stud., 3 1975, 1--25.
  • I. Zang, A smoothing out technique for min-max optimization, Math. Programm., 19 1980, 61--77.
  • A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog., 43 2010, 731--737.
  • A. M. Bagirov, A. Al Nuamiat and N. Sultanova, Hyperbolic smoothing functions for nonsmooth minimization, Optimization, 62 (6) 2013, 759--782.
  • M. Jiang, R. Shen, X. Xu, Z. Meng, Second-order smoothing objective penalty function for constrained optimization problems, Numer. Func. Anal. Opt., 35 2014, 294--309.
  • S. J. Lian Smoothing approximation to $l_1$ exact penalty for inequality constrained optimization, Appl. Math. Comput., 219 2012, 3113--3121.
  • C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 1996, 97--138.
  • X. Chen and W. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imaging Sciences, 3(4) 2010, 765--790.
  • G.W. Weber, I. Batmaz and G. Koksal, CMARS: a new contribution to nonparametric regression with multivariate adaptiveregression splines supported by continuous optimization, Inverse. Probl. Eng., 20(3) 2012, 371--400.
  • P. Taylan, G.-W. Weber, F. Yerlikaya Ozkurt, A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization. TOP 18(2) 2010, 377--395.
  • A. Sahiner, N. Yilmaz and G. Kapusuz, A new global optimization method and applications, Carpathian Math. J., 33(3) 2017, 373-380.
  • A. Sahiner, S. A. Ibrahem, A new global optimization technique by auxiliary function method in a directional search, Optim. Lett. 2018, doi: 10.1007/s11590-018-1315-1
  • J. Lee, D. Skipper, Virtuous smoothing for global optimization, J. Glob. Optim., 69(3) 2017, 677-697.
  • X. Chen, Smoothing methods for nonsmooth, nonconvex minimzation, Math. Program. Ser. B, 134 2012, 71--99.
  • C. Grossmann, Smoothing techniques for exact penalty function methods, Contemporary Mathematics, 658 2016, 249--265.
  • F. G. Vazquez, H. Gunzel and J. J. Ruckmann, On logarithmic smoothing of the maximum function, Ann. Oper. Res., 101 2001, 209--220.
  • H. Th. Jongen and D. Pallaschke, On linearization and continuous selection of functions, Optimization, 19(3) 1988, 343--353.

Year 2018, Volume: 1 Issue: 2, 69 - 74, 21.01.2019

Ahmet Sahiner , Nurullah Yilmaz , Shehab Ahmed Ibrahem

Abstract

References

  • R. T. Rockafellar, R. J-B. Wets R. J-B., Variational Analysis, Springer, Berlin, 1998.
  • F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
  • V. F. Demyanov, A. M. Rubinov, Constructive Nonsmooth Analysis. Verlag Peter Lang, Frankfurt, 1995.
  • B. Akteke-Ozkurt, G. W. Weber, G. Koksal, Optimization of generalized desirability functions under model uncertainty, Optimization, 66 (12) 2017, 2157--2169.
  • A. Sahiner, N. Yilmaz and G. Kapusuz, A novel modeling and smoothing technique in global optimization, J. Ind. Manage. Optim., 15 (1) 2019, 113-130.
  • D. Bertsekas, Nondifferentiable optimization via approximation, Math. Program. Stud., 3 1975, 1--25.
  • I. Zang, A smoothing out technique for min-max optimization, Math. Programm., 19 1980, 61--77.
  • A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog., 43 2010, 731--737.
  • A. M. Bagirov, A. Al Nuamiat and N. Sultanova, Hyperbolic smoothing functions for nonsmooth minimization, Optimization, 62 (6) 2013, 759--782.
  • M. Jiang, R. Shen, X. Xu, Z. Meng, Second-order smoothing objective penalty function for constrained optimization problems, Numer. Func. Anal. Opt., 35 2014, 294--309.
  • S. J. Lian Smoothing approximation to $l_1$ exact penalty for inequality constrained optimization, Appl. Math. Comput., 219 2012, 3113--3121.
  • C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 1996, 97--138.
  • X. Chen and W. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imaging Sciences, 3(4) 2010, 765--790.
  • G.W. Weber, I. Batmaz and G. Koksal, CMARS: a new contribution to nonparametric regression with multivariate adaptiveregression splines supported by continuous optimization, Inverse. Probl. Eng., 20(3) 2012, 371--400.
  • P. Taylan, G.-W. Weber, F. Yerlikaya Ozkurt, A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization. TOP 18(2) 2010, 377--395.
  • A. Sahiner, N. Yilmaz and G. Kapusuz, A new global optimization method and applications, Carpathian Math. J., 33(3) 2017, 373-380.
  • A. Sahiner, S. A. Ibrahem, A new global optimization technique by auxiliary function method in a directional search, Optim. Lett. 2018, doi: 10.1007/s11590-018-1315-1
  • J. Lee, D. Skipper, Virtuous smoothing for global optimization, J. Glob. Optim., 69(3) 2017, 677-697.
  • X. Chen, Smoothing methods for nonsmooth, nonconvex minimzation, Math. Program. Ser. B, 134 2012, 71--99.
  • C. Grossmann, Smoothing techniques for exact penalty function methods, Contemporary Mathematics, 658 2016, 249--265.
  • F. G. Vazquez, H. Gunzel and J. J. Ruckmann, On logarithmic smoothing of the maximum function, Ann. Oper. Res., 101 2001, 209--220.
  • H. Th. Jongen and D. Pallaschke, On linearization and continuous selection of functions, Optimization, 19(3) 1988, 343--353.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ahmet Sahiner 0000-0002-4945-2476

Nurullah Yilmaz 0000-0001-6429-7518

Shehab Ahmed Ibrahem 0000-0003-2428-2703

Publication Date January 21, 2019
Published in Issue Year 2018 Volume: 1 Issue: 2

Cite

APA Sahiner, A., Yilmaz, N., & Ibrahem, S. A. (2019). Smoothing Approximations to Non-smooth Functions. Journal of Multidisciplinary Modeling and Optimization, 1(2), 69-74.
AMA Sahiner A, Yilmaz N, Ibrahem SA. Smoothing Approximations to Non-smooth Functions. jmmo. January 2019;1(2):69-74.
Chicago Sahiner, Ahmet, Nurullah Yilmaz, and Shehab Ahmed Ibrahem. “Smoothing Approximations to Non-Smooth Functions”. Journal of Multidisciplinary Modeling and Optimization 1, no. 2 (January 2019): 69-74.
EndNote Sahiner A, Yilmaz N, Ibrahem SA (January 1, 2019) Smoothing Approximations to Non-smooth Functions. Journal of Multidisciplinary Modeling and Optimization 1 2 69–74.
IEEE A. Sahiner, N. Yilmaz, and S. A. Ibrahem, “Smoothing Approximations to Non-smooth Functions”, jmmo, vol. 1, no. 2, pp. 69–74, 2019.
ISNAD Sahiner, Ahmet et al. “Smoothing Approximations to Non-Smooth Functions”. Journal of Multidisciplinary Modeling and Optimization 1/2 (January 2019), 69-74.
JAMA Sahiner A, Yilmaz N, Ibrahem SA. Smoothing Approximations to Non-smooth Functions. jmmo. 2019;1:69–74.
MLA Sahiner, Ahmet et al. “Smoothing Approximations to Non-Smooth Functions”. Journal of Multidisciplinary Modeling and Optimization, vol. 1, no. 2, 2019, pp. 69-74.
Vancouver Sahiner A, Yilmaz N, Ibrahem SA. Smoothing Approximations to Non-smooth Functions. jmmo. 2019;1(2):69-74.
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