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.
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.
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.
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.