Smoothing Approximations to Non-smooth Functions
Yıl 2018,
Cilt: 1 Sayı: 2, 69 - 74, 21.01.2019
Ahmet Sahiner
,
Nurullah Yilmaz
,
Shehab Ahmed Ibrahem
Öz
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.
Kaynakça
- 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.
Yıl 2018,
Cilt: 1 Sayı: 2, 69 - 74, 21.01.2019
Ahmet Sahiner
,
Nurullah Yilmaz
,
Shehab Ahmed Ibrahem
Kaynakça
- 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.