A novel modeling and smoothing technique in global optimization. (English) Zbl 1415.90093
Summary: 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.
MSC:
| 90C26 | Nonconvex programming, global optimization |
| 65D05 | Numerical interpolation |
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 97M10 | Modeling and interdisciplinarity (aspects of mathematics education) |
Software:
CMARSReferences:
| [1] | B. Belkhatir; A. Zidna, Construction of flexible blending parametric surfaces via curves, Math. Comput. Simulat., 79, 3599-3608 (2009) · Zbl 1178.65016 · doi:10.1016/j.matcom.2009.04.015 |
| [2] | X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134, 71-99 (2012) · Zbl 1266.90145 · doi:10.1007/s10107-012-0569-0 |
| [3] | J. Cheng; X. S. Gao, Constructing blending surfaces for two arbitrary surfaces, MM Research Preprints, 22, 14-28 (2003) |
| [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; A. I. Ahmet, A new filled function method applied to unconstrained global optimization, Appl. Math. Comput., 273, 1246-1256 (2016) · Zbl 1410.90165 · 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. · Zbl 1003.68179 |
| [8] | R. P. Ge, A filled function method for finding a global minimizer of a function of several variables, Math. Program., 46, 191-204 (1990) · Zbl 0694.90083 · 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, 1-9 (1987) · Zbl 0692.90087 |
| [10] | A. Griewank; A. Walther, First-and second-order optimality conditions for piecewise smooth objective functions, Optim. Method Softw., 31, 904-930 (2016) · Zbl 1385.90026 · doi:10.1080/10556788.2016.1189549 |
| [11] | T. Gu; S. Ji; S. Lin; T. Luo, Curve and surface reconstruction method for measurement data, Measurement, 78, 278-282 (2016) · doi:10.1016/j.measurement.2015.10.011 |
| [12] | K. A. Guzzetti; A. B. Brizuela; E. Romano; 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, 171-179 (2013) · doi:10.1016/j.molstruc.2013.04.016 |
| [13] | E. Hartmann, Blending an implicit with a parametric surface, Comput. Aided Geom. D., 12, 825-835 (1995) · Zbl 0875.65032 · doi:10.1016/0167-8396(95)00002-1 |
| [14] | W. Kohn; L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140, A1133-A1138 (1965) · doi:10.1103/PhysRev.140.A1133 |
| [15] | A. V. Levy; A. Montalvo, The tunneling algorithm for the global minimization of functions, SIAM Sci. Stat. Comput., 6, 15-29 (1985) · Zbl 0601.65050 · doi:10.1137/0906002 |
| [16] | S. Ma; Y. Yang; H. Liu, A parameter free filled function for unconstrained global optimization, Appl. Math. Comput., 215, 3610-3619 (2010) · Zbl 1192.65081 · doi:10.1016/j.amc.2009.10.057 |
| [17] | A. Mazroui; D. Sbibih; A. Tijini, A simple method for smoothing functions and compressing Hermite data, Adv. Comput. Math., 23, 279-297 (2005) · Zbl 1070.65011 · doi:10.1007/s10444-004-1783-y |
| [18] | A. Mazroui; H. Mraoui; D. Sbibih; A. Tijini, A simple method for smoothing functions and compressing Hermite data, BIT Numerical Mathematics, 47, 613-635 (2007) · Zbl 1123.41002 · doi:10.1007/s10543-007-0139-7 |
| [19] | C. K. Ng; D. Li; L. S. Zhang, Global descent method for global optimization, SIAM J. Optim., 20, 3161-3184 (2010) · Zbl 1208.49038 · doi:10.1137/090749815 |
| [20] | I. Nowak; J. Smolka; A. J. Nowak, Application of Bezier surfaces to the 3-D inverse geometry problem in continuous casting, Inverse Probl. Sci. Eng., 19, 75-86 (2011) · Zbl 1452.76166 |
| [21] | A. Ozmen; G. W. Weber; I. Batmaz; E. Kropat, RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set, Commun. Nonlinear Sci. Numer. Simulat., 16, 4780-4787 (2011) · Zbl 1416.65169 · 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; N. Yilmaz, Completed optimised structure of threonine molecule by fuzzy logic modelling, Z. Naturforsh. A, 71, 381-386 (2016) · doi:10.1515/zna-2015-0424 |
| [24] | A. Sahiner; N. Yilmaz; G. Kapusuz, A descent global optimization method based on smoothing techniques via Bezier curves, Carpathian J. Math., 33, 373-380 (2017) · Zbl 1399.90214 |
| [25] | Y. D. Sergeyev; D. E. Kvasov, A deterministic global optimization using smooth diagonal auxiliary functions, Commun. Nonlinear Sci. Numer. Simulat., 21, 99-111 (2015) · Zbl 1329.90112 · doi:10.1016/j.cnsns.2014.08.026 |
| [26] | P. Venkataraman, Solution of inverse ODE using Bezier functions, Inverse Probl. Sci. Eng., 19, 529-549 (2011) · Zbl 1220.65097 · doi:10.1080/17415977.2010.531465 |
| [27] | G. W. Weber; I. Batmaz; G. Koksal; P. Taylan; F. Yerlikaya-Ozkurt, CMARS: A new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization, Inverse. Probl. Eng., 20, 371-400 (2012) · Zbl 1254.65020 · doi:10.1080/17415977.2011.624770 |
| [28] | Z. Y. Wu; D. Li; L. S. Zhang, Global descent methods for unconstrained global optimization, J. Glob. Optim., 50, 379-396 (2011) · Zbl 1228.90089 · doi:10.1007/s10898-010-9587-8 |
| [29] | H. Wu; P. Zhang; G. H. Lin, Smoothing approximations for some piecewise smooth functions, J. Oper. Res. Soc. China, 3, 317-329 (2015) · Zbl 1326.26010 · doi:10.1007/s40305-015-0091-1 |
| [30] | Y. T. Xu; Y. Zhang; S. G. Wang, A modified tunneling function method for non-smooth global optimization and its application in artificial neural network, Appl. Math. Model., 39, 6438-6450 (2015) · Zbl 1443.90332 · doi:10.1016/j.apm.2015.01.059 |
| [31] | X. Ye; Y. Liang; H. Nowacki, Geometric continuity between adjacent Bézier patches and their constructions, Comput. Aided Geom. D., 13, 521-548 (1996) · Zbl 0875.68870 · 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. · Zbl 1414.90334 |
| [33] | N. Yilmaz and A. Sahiner, New global optimization method for non-smooth unconstrained continuous optimization AIP Conference Proceedings, 1863 (2017), 250002. |
| [34] | J. Zilinskas, Branch and bound with simplicial partitions for global optimization, Math. Model. Anal., 13, 145-159 (2008) · Zbl 1146.49029 · doi:10.3846/1392-6292.2008.13.145-159 |
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