
-
Previous Article
Mechanism design in project procurement auctions with cost uncertainty and failure risk
- JIMO Home
- This Issue
-
Next Article
Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank
A novel modeling and smoothing technique in global optimization
Suleyman Demirel University, Department of Mathematics, Isparta, 32100, Turkey |
In this paper, we introduce a new methodology for modeling of the given data and finding the global optimum value of the model function. First, a new surface blending technique is offered by using Bezier curves and a smooth objective function is obtained with the help of this technique. Second, a new global optimization method followed by an adapted algorithm is presented to reach the global minimizer of the objective function. As an application of this new methodology, we consider energy conformation problem in Physical Chemistry as a very important real-world problem.
References:
[1] |
B. Belkhatir and A. Zidna,
Construction of flexible blending parametric surfaces via curves, Math. Comput. Simulat., 79 (2009), 3599-3608.
doi: 10.1016/j.matcom.2009.04.015. |
[2] |
X. Chen,
Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.
doi: 10.1007/s10107-012-0569-0. |
[3] |
J. Cheng and X. S. Gao,
Constructing blending surfaces for two arbitrary surfaces, MM Research Preprints, 22 (2003), 14-28.
|
[4] |
R. M. C. Dawson, D. C. Elliot and K. M. Jones,
Data for Biochemical Research, Clarendon Press, Oxford, 1985. |
[5] |
T. M. El-Gindy, M. S. Salim and A. I. Ahmet,
A new filled function method applied to unconstrained global optimization, Appl. Math. Comput., 273 (2016), 1246-1256.
doi: 10.1016/j.amc.2015.08.091. |
[6] |
G. E. Farin,
Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann, San Fransico, 2002. |
[7] |
G. E. Farin, J. Hoschek and M. S. Kim,
Handbook of Computer Aided Geometric Design, Elsevier, Amsterdam, 2002. |
[8] |
R. P. Ge,
A filled function method for finding a global minimizer of a function of several variables, Math. Program., 46 (1990), 191-204.
doi: 10.1007/BF01585737. |
[9] |
R. P. Ge,
The theory of filled function method for finding global minimizers of nonlinearly constrained minimization problems, J. Comput. Math., 5 (1987), 1-9.
|
[10] |
A. Griewank and A. Walther,
First-and second-order optimality conditions for piecewise smooth objective functions, Optim. Method Softw., 31 (2016), 904-930.
doi: 10.1080/10556788.2016.1189549. |
[11] |
T. Gu, S. Ji, S. Lin and T. Luo,
Curve and surface reconstruction method for measurement data, Measurement, 78 (2016), 278-282.
doi: 10.1016/j.measurement.2015.10.011. |
[12] |
K. A. Guzzetti, A. B. Brizuela, E. Romano and S. A. Brandán,
Structural and vibrational study on zwitterions of l-threonine in aqueous phase using the FT-Raman and SCRF calculations, Mol. Struct., 1045 (2013), 171-179.
doi: 10.1016/j.molstruc.2013.04.016. |
[13] |
E. Hartmann,
Blending an implicit with a parametric surface, Comput. Aided Geom. D., 12 (1995), 825-835.
doi: 10.1016/0167-8396(95)00002-1. |
[14] |
W. Kohn and L. J. Sham,
Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), A1133-A1138.
doi: 10.1103/PhysRev.140.A1133. |
[15] |
A. V. Levy and A. Montalvo,
The tunneling algorithm for the global minimization of functions, SIAM Sci. Stat. Comput., 6 (1985), 15-29.
doi: 10.1137/0906002. |
[16] |
S. Ma, Y. Yang and H. Liu,
A parameter free filled function for unconstrained global optimization, Appl. Math. Comput., 215 (2010), 3610-3619.
doi: 10.1016/j.amc.2009.10.057. |
[17] |
A. Mazroui, D. Sbibih and A. Tijini,
A simple method for smoothing functions and compressing Hermite data, Adv. Comput. Math., 23 (2005), 279-297.
doi: 10.1007/s10444-004-1783-y. |
[18] |
A. Mazroui, H. Mraoui, D. Sbibih and A. Tijini,
A simple method for smoothing functions and compressing Hermite data, BIT Numerical Mathematics, 47 (2007), 613-635.
doi: 10.1007/s10543-007-0139-7. |
[19] |
C. K. Ng, D. Li and L. S. Zhang,
Global descent method for global optimization, SIAM J. Optim., 20 (2010), 3161-3184.
doi: 10.1137/090749815. |
[20] |
I. Nowak, J. Smolka and A. J. Nowak,
Application of Bezier surfaces to the 3-D inverse geometry problem in continuous casting, Inverse Probl. Sci. Eng., 19 (2011), 75-86.
|
[21] |
A. Ozmen, G. W. Weber, I. Batmaz and E. Kropat,
RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 4780-4787.
doi: 10.1016/j.cnsns.2011.04.001. |
[22] |
R. G. Parr and W. G. Yang,
Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. |
[23] |
A. Sahiner, F. Ucun, G. Kapusuz and N. Yilmaz,
Completed optimised structure of threonine molecule by fuzzy logic modelling, Z. Naturforsh. A, 71 (2016), 381-386.
doi: 10.1515/zna-2015-0424. |
[24] |
A. Sahiner, N. Yilmaz and G. Kapusuz,
A descent global optimization method based on smoothing techniques via Bezier curves, Carpathian J. Math., 33 (2017), 373-380.
|
[25] |
Y. D. Sergeyev and D. E. Kvasov,
A deterministic global optimization using smooth diagonal auxiliary functions, Commun. Nonlinear Sci. Numer. Simulat., 21 (2015), 99-111.
doi: 10.1016/j.cnsns.2014.08.026. |
[26] |
P. Venkataraman,
Solution of inverse ODE using Bezier functions, Inverse Probl. Sci. Eng., 19 (2011), 529-549.
doi: 10.1080/17415977.2010.531465. |
[27] |
G. W. Weber, I. Batmaz, G. Koksal, P. Taylan and F. Yerlikaya-Ozkurt,
CMARS: A new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization, Inverse. Probl. Eng., 20 (2012), 371-400.
doi: 10.1080/17415977.2011.624770. |
[28] |
Z. Y. Wu, D. Li and L. S. Zhang,
Global descent methods for unconstrained global optimization, J. Glob. Optim., 50 (2011), 379-396.
doi: 10.1007/s10898-010-9587-8. |
[29] |
H. Wu, P. Zhang and G. H. Lin,
Smoothing approximations for some piecewise smooth functions, J. Oper. Res. Soc. China, 3 (2015), 317-329.
doi: 10.1007/s40305-015-0091-1. |
[30] |
Y. T. Xu, Y. Zhang and S. G. Wang,
A modified tunneling function method for non-smooth global optimization and its application in artificial neural network, Appl. Math. Model., 39 (2015), 6438-6450.
doi: 10.1016/j.apm.2015.01.059. |
[31] |
X. Ye, Y. Liang and H. Nowacki,
Geometric continuity between adjacent Bézier patches and their constructions, Comput. Aided Geom. D., 13 (1996), 521-548.
doi: 10.1016/0167-8396(95)00043-7. |
[32] |
N. Yilmaz and A. Sahiner, A new smoothing approximation to piecewise smooth functions and applications,
International Conference on Analysis and Application, 1 (2016), p226. |
[33] |
N. Yilmaz and A. Sahiner, New global optimization method for non-smooth unconstrained continuous optimization AIP Conference Proceedings, 1863 (2017), 250002.
doi: 10.1063/1.4992410. |
[34] |
J. Zilinskas,
Branch and bound with simplicial partitions for global optimization, Math. Model. Anal., 13 (2008), 145-159.
doi: 10.3846/1392-6292.2008.13.145-159. |
show all references
References:
[1] |
B. Belkhatir and A. Zidna,
Construction of flexible blending parametric surfaces via curves, Math. Comput. Simulat., 79 (2009), 3599-3608.
doi: 10.1016/j.matcom.2009.04.015. |
[2] |
X. Chen,
Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.
doi: 10.1007/s10107-012-0569-0. |
[3] |
J. Cheng and X. S. Gao,
Constructing blending surfaces for two arbitrary surfaces, MM Research Preprints, 22 (2003), 14-28.
|
[4] |
R. M. C. Dawson, D. C. Elliot and K. M. Jones,
Data for Biochemical Research, Clarendon Press, Oxford, 1985. |
[5] |
T. M. El-Gindy, M. S. Salim and A. I. Ahmet,
A new filled function method applied to unconstrained global optimization, Appl. Math. Comput., 273 (2016), 1246-1256.
doi: 10.1016/j.amc.2015.08.091. |
[6] |
G. E. Farin,
Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann, San Fransico, 2002. |
[7] |
G. E. Farin, J. Hoschek and M. S. Kim,
Handbook of Computer Aided Geometric Design, Elsevier, Amsterdam, 2002. |
[8] |
R. P. Ge,
A filled function method for finding a global minimizer of a function of several variables, Math. Program., 46 (1990), 191-204.
doi: 10.1007/BF01585737. |
[9] |
R. P. Ge,
The theory of filled function method for finding global minimizers of nonlinearly constrained minimization problems, J. Comput. Math., 5 (1987), 1-9.
|
[10] |
A. Griewank and A. Walther,
First-and second-order optimality conditions for piecewise smooth objective functions, Optim. Method Softw., 31 (2016), 904-930.
doi: 10.1080/10556788.2016.1189549. |
[11] |
T. Gu, S. Ji, S. Lin and T. Luo,
Curve and surface reconstruction method for measurement data, Measurement, 78 (2016), 278-282.
doi: 10.1016/j.measurement.2015.10.011. |
[12] |
K. A. Guzzetti, A. B. Brizuela, E. Romano and S. A. Brandán,
Structural and vibrational study on zwitterions of l-threonine in aqueous phase using the FT-Raman and SCRF calculations, Mol. Struct., 1045 (2013), 171-179.
doi: 10.1016/j.molstruc.2013.04.016. |
[13] |
E. Hartmann,
Blending an implicit with a parametric surface, Comput. Aided Geom. D., 12 (1995), 825-835.
doi: 10.1016/0167-8396(95)00002-1. |
[14] |
W. Kohn and L. J. Sham,
Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), A1133-A1138.
doi: 10.1103/PhysRev.140.A1133. |
[15] |
A. V. Levy and A. Montalvo,
The tunneling algorithm for the global minimization of functions, SIAM Sci. Stat. Comput., 6 (1985), 15-29.
doi: 10.1137/0906002. |
[16] |
S. Ma, Y. Yang and H. Liu,
A parameter free filled function for unconstrained global optimization, Appl. Math. Comput., 215 (2010), 3610-3619.
doi: 10.1016/j.amc.2009.10.057. |
[17] |
A. Mazroui, D. Sbibih and A. Tijini,
A simple method for smoothing functions and compressing Hermite data, Adv. Comput. Math., 23 (2005), 279-297.
doi: 10.1007/s10444-004-1783-y. |
[18] |
A. Mazroui, H. Mraoui, D. Sbibih and A. Tijini,
A simple method for smoothing functions and compressing Hermite data, BIT Numerical Mathematics, 47 (2007), 613-635.
doi: 10.1007/s10543-007-0139-7. |
[19] |
C. K. Ng, D. Li and L. S. Zhang,
Global descent method for global optimization, SIAM J. Optim., 20 (2010), 3161-3184.
doi: 10.1137/090749815. |
[20] |
I. Nowak, J. Smolka and A. J. Nowak,
Application of Bezier surfaces to the 3-D inverse geometry problem in continuous casting, Inverse Probl. Sci. Eng., 19 (2011), 75-86.
|
[21] |
A. Ozmen, G. W. Weber, I. Batmaz and E. Kropat,
RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 4780-4787.
doi: 10.1016/j.cnsns.2011.04.001. |
[22] |
R. G. Parr and W. G. Yang,
Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. |
[23] |
A. Sahiner, F. Ucun, G. Kapusuz and N. Yilmaz,
Completed optimised structure of threonine molecule by fuzzy logic modelling, Z. Naturforsh. A, 71 (2016), 381-386.
doi: 10.1515/zna-2015-0424. |
[24] |
A. Sahiner, N. Yilmaz and G. Kapusuz,
A descent global optimization method based on smoothing techniques via Bezier curves, Carpathian J. Math., 33 (2017), 373-380.
|
[25] |
Y. D. Sergeyev and D. E. Kvasov,
A deterministic global optimization using smooth diagonal auxiliary functions, Commun. Nonlinear Sci. Numer. Simulat., 21 (2015), 99-111.
doi: 10.1016/j.cnsns.2014.08.026. |
[26] |
P. Venkataraman,
Solution of inverse ODE using Bezier functions, Inverse Probl. Sci. Eng., 19 (2011), 529-549.
doi: 10.1080/17415977.2010.531465. |
[27] |
G. W. Weber, I. Batmaz, G. Koksal, P. Taylan and F. Yerlikaya-Ozkurt,
CMARS: A new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization, Inverse. Probl. Eng., 20 (2012), 371-400.
doi: 10.1080/17415977.2011.624770. |
[28] |
Z. Y. Wu, D. Li and L. S. Zhang,
Global descent methods for unconstrained global optimization, J. Glob. Optim., 50 (2011), 379-396.
doi: 10.1007/s10898-010-9587-8. |
[29] |
H. Wu, P. Zhang and G. H. Lin,
Smoothing approximations for some piecewise smooth functions, J. Oper. Res. Soc. China, 3 (2015), 317-329.
doi: 10.1007/s40305-015-0091-1. |
[30] |
Y. T. Xu, Y. Zhang and S. G. Wang,
A modified tunneling function method for non-smooth global optimization and its application in artificial neural network, Appl. Math. Model., 39 (2015), 6438-6450.
doi: 10.1016/j.apm.2015.01.059. |
[31] |
X. Ye, Y. Liang and H. Nowacki,
Geometric continuity between adjacent Bézier patches and their constructions, Comput. Aided Geom. D., 13 (1996), 521-548.
doi: 10.1016/0167-8396(95)00043-7. |
[32] |
N. Yilmaz and A. Sahiner, A new smoothing approximation to piecewise smooth functions and applications,
International Conference on Analysis and Application, 1 (2016), p226. |
[33] |
N. Yilmaz and A. Sahiner, New global optimization method for non-smooth unconstrained continuous optimization AIP Conference Proceedings, 1863 (2017), 250002.
doi: 10.1063/1.4992410. |
[34] |
J. Zilinskas,
Branch and bound with simplicial partitions for global optimization, Math. Model. Anal., 13 (2008), 145-159.
doi: 10.3846/1392-6292.2008.13.145-159. |



Problem No. | Function Name | Dimension | Region | Optimum value |
1 | Two dimensional function | |||
2 | Two dimensional function | |||
3 | Two dimensional function | |||
4 | 3-hump function | |||
5 | 6-hump function | |
||
6 | Treccani function | | ||
7 | Goldstein-Price function | | ||
8 | Shubert function | |||
9 | Rastrigin function | |||
10 | Branin function | |||
11 | (S5) Shekel function | |||
12 | (S7) Shekel function | |||
13 | (S10) Shekel function | |||
14, 15, 16, 17 | Sin-square I function | | |
|
18, 19, 20, 21 | Sin-square I function | |
Problem No. | Function Name | Dimension | Region | Optimum value |
1 | Two dimensional function | |||
2 | Two dimensional function | |||
3 | Two dimensional function | |||
4 | 3-hump function | |||
5 | 6-hump function | |
||
6 | Treccani function | | ||
7 | Goldstein-Price function | | ||
8 | Shubert function | |||
9 | Rastrigin function | |||
10 | Branin function | |||
11 | (S5) Shekel function | |||
12 | (S7) Shekel function | |||
13 | (S10) Shekel function | |||
14, 15, 16, 17 | Sin-square I function | | |
|
18, 19, 20, 21 | Sin-square I function | |
Problem No. | n | iter-m | f.eval-m | f-mean | f-best | SR |
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
6 | ||||||
7 | ||||||
8 | ||||||
9 | ||||||
10 | ||||||
11 | ||||||
12 | ||||||
13 | ||||||
14 | ||||||
15 | ||||||
16 | ||||||
17 | ||||||
18 | ||||||
19 | ||||||
20 | ||||||
21 |
Problem No. | n | iter-m | f.eval-m | f-mean | f-best | SR |
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
6 | ||||||
7 | ||||||
8 | ||||||
9 | ||||||
10 | ||||||
11 | ||||||
12 | ||||||
13 | ||||||
14 | ||||||
15 | ||||||
16 | ||||||
17 | ||||||
18 | ||||||
19 | ||||||
20 | ||||||
21 |
No | n | Our Method | Ma et. al [16] | El-Gindy et. al [5] | |||||
iter-m | f.eval-m | iter-m | f.eval-m | iter-m | f.eval-m | ||||
1 | |||||||||
2 | |||||||||
3 | |||||||||
4 | |||||||||
5 | |||||||||
6 | |||||||||
7 | |||||||||
8 | |||||||||
9 | |||||||||
10 | |||||||||
14 | |||||||||
15 | |||||||||
16 | |||||||||
17 | |||||||||
18 | |||||||||
19 | |||||||||
20 | |||||||||
21 |
No | n | Our Method | Ma et. al [16] | El-Gindy et. al [5] | |||||
iter-m | f.eval-m | iter-m | f.eval-m | iter-m | f.eval-m | ||||
1 | |||||||||
2 | |||||||||
3 | |||||||||
4 | |||||||||
5 | |||||||||
6 | |||||||||
7 | |||||||||
8 | |||||||||
9 | |||||||||
10 | |||||||||
14 | |||||||||
15 | |||||||||
16 | |||||||||
17 | |||||||||
18 | |||||||||
19 | |||||||||
20 | |||||||||
21 |
1 | (160.0000,280.0000) | ||||
2 | |||||
3 |
1 | (160.0000,280.0000) | ||||
2 | |||||
3 |
[1] |
Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004 |
[2] |
Lingshuang Kong, Changjun Yu, Kok Lay Teo, Chunhua Yang. Robust real-time optimization for blending operation of alumina production. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1149-1167. doi: 10.3934/jimo.2016066 |
[3] |
Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial and Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533 |
[4] |
Karol Mikula, Mariana Remešíková, Peter Novysedlák. Truss structure design using a length-oriented surface remeshing technique. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 933-951. doi: 10.3934/dcdss.2015.8.933 |
[5] |
Yu Chen, Yonggang Li, Bei Sun, Chunhua Yang, Hongqiu Zhu. Multi-objective chance-constrained blending optimization of zinc smelter under stochastic uncertainty. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021169 |
[6] |
Valentin R. Koch, Yves Lucet. A note on: Spline technique for modeling roadway profile to minimize earthwork cost. Journal of Industrial and Management Optimization, 2010, 6 (2) : 393-400. doi: 10.3934/jimo.2010.6.393 |
[7] |
Ahmad A. Moreb. Spline technique for modeling roadway profile to minimize earthwork cost. Journal of Industrial and Management Optimization, 2009, 5 (2) : 275-283. doi: 10.3934/jimo.2009.5.275 |
[8] |
Vadim Azhmyakov. An approach to controlled mechanical systems based on the multiobjective optimization technique. Journal of Industrial and Management Optimization, 2008, 4 (4) : 697-712. doi: 10.3934/jimo.2008.4.697 |
[9] |
Xi Zhu, Changjun Yu, Kok Lay Teo. A new switching time optimization technique for multi-switching systems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022067 |
[10] |
Jutamas Kerdkaew, Rabian Wangkeeree, Rattanaporn Wangkeeree. Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 93-107. doi: 10.3934/naco.2021053 |
[11] |
Ahmet Sahiner, Gulden Kapusuz, Nurullah Yilmaz. A new smoothing approach to exact penalty functions for inequality constrained optimization problems. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 161-173. doi: 10.3934/naco.2016006 |
[12] |
X. X. Huang, Xiaoqi Yang, K. L. Teo. A smoothing scheme for optimization problems with Max-Min constraints. Journal of Industrial and Management Optimization, 2007, 3 (2) : 209-222. doi: 10.3934/jimo.2007.3.209 |
[13] |
Anna Chiara Lai, Monica Motta. Stabilizability in optimization problems with unbounded data. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2447-2474. doi: 10.3934/dcds.2020371 |
[14] |
Colette Calmelet, Diane Sepich. Surface tension and modeling of cellular intercalation during zebrafish gastrulation. Mathematical Biosciences & Engineering, 2010, 7 (2) : 259-275. doi: 10.3934/mbe.2010.7.259 |
[15] |
Emre Kiliç, Mehmet Çayören, Ali Yapar, Íbrahim Akduman. Reconstruction of perfectly conducting rough surfaces by the use of inhomogeneous surface impedance modeling. Inverse Problems and Imaging, 2009, 3 (2) : 295-307. doi: 10.3934/ipi.2009.3.295 |
[16] |
Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems and Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033 |
[17] |
Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems and Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341 |
[18] |
Runqin Hao, Guanwen Zhang, Dong Li, Jie Zhang. Data modeling analysis on removal efficiency of hexavalent chromium. Mathematical Foundations of Computing, 2019, 2 (3) : 203-213. doi: 10.3934/mfc.2019014 |
[19] |
Martha Garlick, James Powell, David Eyre, Thomas Robbins. Mathematically modeling PCR: An asymptotic approximation with potential for optimization. Mathematical Biosciences & Engineering, 2010, 7 (2) : 363-384. doi: 10.3934/mbe.2010.7.363 |
[20] |
Michael Herty. Modeling, simulation and optimization of gas networks with compressors. Networks and Heterogeneous Media, 2007, 2 (1) : 81-97. doi: 10.3934/nhm.2007.2.81 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]