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January  2019, 15(1): 113-130. doi: 10.3934/jimo.2018035

A novel modeling and smoothing technique in global optimization

Ahmet Sahiner , , Nurullah Yilmaz and  Gulden Kapusuz

Suleyman Demirel University, Department of Mathematics, Isparta, 32100, Turkey

* Corresponding author: ahmetsahiner@sdu.edu.tr

Received  April 2017 Revised  January 2018 Published  January 2019 Early access  April 2018

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.

Keywords: Global optimization, blending surface, data modeling, smoothing technique.
Mathematics Subject Classification: Primary: 90C26; Secondary: 65D05, 65D17, 97M10.
Citation: Ahmet Sahiner, Nurullah Yilmaz, Gulden Kapusuz. A novel modeling and smoothing technique in global optimization. Journal of Industrial & Management Optimization, 2019, 15 (1) : 113-130. doi: 10.3934/jimo.2018035
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.  Google Scholar

[2]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[3]

J. Cheng and X. S. Gao, Constructing blending surfaces for two arbitrary surfaces, MM Research Preprints, 22 (2003), 14-28.   Google Scholar

[4]

R. M. C. Dawson, D. C. Elliot and K. M. Jones, Data for Biochemical Research, Clarendon Press, Oxford, 1985. Google Scholar

[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.  Google Scholar

[6]

G. E. Farin, Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann, San Fransico, 2002. Google Scholar

[7]

G. E. Farin, J. Hoschek and M. S. Kim, Handbook of Computer Aided Geometric Design, Elsevier, Amsterdam, 2002.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

R. G. Parr and W. G. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

P. Venkataraman, Solution of inverse ODE using Bezier functions, Inverse Probl. Sci. Eng., 19 (2011), 529-549.  doi: 10.1080/17415977.2010.531465.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[3]

J. Cheng and X. S. Gao, Constructing blending surfaces for two arbitrary surfaces, MM Research Preprints, 22 (2003), 14-28.   Google Scholar

[4]

R. M. C. Dawson, D. C. Elliot and K. M. Jones, Data for Biochemical Research, Clarendon Press, Oxford, 1985. Google Scholar

[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.  Google Scholar

[6]

G. E. Farin, Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann, San Fransico, 2002. Google Scholar

[7]

G. E. Farin, J. Hoschek and M. S. Kim, Handbook of Computer Aided Geometric Design, Elsevier, Amsterdam, 2002.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

R. G. Parr and W. G. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

P. Venkataraman, Solution of inverse ODE using Bezier functions, Inverse Probl. Sci. Eng., 19 (2011), 529-549.  doi: 10.1080/17415977.2010.531465.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The subregions of $\Omega = [0,360]\times[0,360]$
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Figure 2.  Constructed Bezier surfaces on the subregions $-438$ was taken as zero to remove the complexity
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Figure 3.  The graph of the function $\tilde{f}(x, y, \varepsilon, \delta)$ which is constructed by blending Bezier surfaces
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Table 1.  The list of test problems
Problem No.Function NameDimension $n$RegionOptimum value
1Two dimensional function $c=0.05$$2$$[-3, 3]^2$$0$
2Two dimensional function $c=0.2$$2$$[-3, 3]^2$$0$
3Two dimensional function $c=0.5$$2$$[-3, 3]^2$$0$
43-hump function$2$$[-3, 3]^2$$0$
56-hump function$2$$[-3, 3]^2$ $-1.0316$
6Treccani function $2$$[-3, 3]^2$$0$
7Goldstein-Price function $2$$[-3, 3]^2$$3.0000$
8Shubert function$2$$[-10, 10]^2$$-186.73091$
9Rastrigin function$2$$[-3, 3]^2$$-2.0000$
10Branin function$2$$[-5, 10]\times[10],[15]$$0.3979$
11(S5) Shekel function$4$$[0, 10]^4$$-10.1532$
12(S7) Shekel function$4$$[0, 10]^4$$-10.4029$
13(S10) Shekel function$4$$[0, 10]^4$$-10.5364$
14, 15, 16, 17Sin-square I function $2, 3, 5, 7$$[-10, 10]^n$ $0$
18, 19, 20, 21Sin-square I function$10, 20, 30, 50$$[-10, 10]^n$ $0$
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Problem No.Function NameDimension $n$RegionOptimum value
1Two dimensional function $c=0.05$$2$$[-3, 3]^2$$0$
2Two dimensional function $c=0.2$$2$$[-3, 3]^2$$0$
3Two dimensional function $c=0.5$$2$$[-3, 3]^2$$0$
43-hump function$2$$[-3, 3]^2$$0$
56-hump function$2$$[-3, 3]^2$ $-1.0316$
6Treccani function $2$$[-3, 3]^2$$0$
7Goldstein-Price function $2$$[-3, 3]^2$$3.0000$
8Shubert function$2$$[-10, 10]^2$$-186.73091$
9Rastrigin function$2$$[-3, 3]^2$$-2.0000$
10Branin function$2$$[-5, 10]\times[10],[15]$$0.3979$
11(S5) Shekel function$4$$[0, 10]^4$$-10.1532$
12(S7) Shekel function$4$$[0, 10]^4$$-10.4029$
13(S10) Shekel function$4$$[0, 10]^4$$-10.5364$
14, 15, 16, 17Sin-square I function $2, 3, 5, 7$$[-10, 10]^n$ $0$
18, 19, 20, 21Sin-square I function$10, 20, 30, 50$$[-10, 10]^n$ $0$
Table 2.  The numerical results of our method
Problem No.niter-mf.eval-mf-meanf-bestSR
1$2$$1.50004$$214$$5.9087e-15$$2.6630e-154$$8/10$
2$2$$1.1250$$290.6250$$7.5789e-15$$3.4336e-16$$8/10$
3$2$$1.7500$$414.2857$$4.0814e-15$$4.7243e-16$$8/10$
4$2$$1.4000$$411$$4.8635e-15$$2.8802e-16$$10/10$
5$2$$1.5000$$234$$-1.0316$$-1.0316$$10/10$
6$2$$1.0000$$216.5000$$5.5963e-14$$1.6477e-15$$10/10$
7$2$$1.2222$$487.8889$$3.0000$$3.0000$$9/10$
8$2$$2.7000$$813.5000$$-186.7309$$-186.7309$$10/10$
9$2$$3.4000$$501$$-2.0000$$-2.0000$$10/10$
10$2$$1.0000$$222.3000$$0.3979$$0.3979$$10/10$
11$4$$1.6667$$1001$$-10.1532$$-10.1532$$9/10$
12$4$$1.7500$$1365.1000$$-10.4029$$-10.4029$$8/10$
13$4$$1.2857$$1412$$-10.5321$$-10.5321$$7/10$
14$2$$2.7500$$743.2500$$9.6751e-15$$9.4192e-15$$8/10$
15$3$$1.9000$$3027$$1.3445e-14$$5.6998e-15$$10/10$
16$5$$1.8000$$4999.3$$1.8351e-13$$3.7007e-15$$10/10$
17$7$$1.7500$$8171$$1.7275e-14$$1.3790e-14$$8/10$
18$10$$2.7778$$8895.4$$4.3639e-13$$3.0992e-14$$9/10$
19$20$$2.7143$$18242$$2.2066e-12$$3.0016e-13$$7/10$
20$30$$3.5000$$43232$$6.9372e-12$$1.7361e-12$$6/10$
21$50$$2.5000$$83243$$7.0303e-12$$9.8531e-13$$6/10$
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Problem No.niter-mf.eval-mf-meanf-bestSR
1$2$$1.50004$$214$$5.9087e-15$$2.6630e-154$$8/10$
2$2$$1.1250$$290.6250$$7.5789e-15$$3.4336e-16$$8/10$
3$2$$1.7500$$414.2857$$4.0814e-15$$4.7243e-16$$8/10$
4$2$$1.4000$$411$$4.8635e-15$$2.8802e-16$$10/10$
5$2$$1.5000$$234$$-1.0316$$-1.0316$$10/10$
6$2$$1.0000$$216.5000$$5.5963e-14$$1.6477e-15$$10/10$
7$2$$1.2222$$487.8889$$3.0000$$3.0000$$9/10$
8$2$$2.7000$$813.5000$$-186.7309$$-186.7309$$10/10$
9$2$$3.4000$$501$$-2.0000$$-2.0000$$10/10$
10$2$$1.0000$$222.3000$$0.3979$$0.3979$$10/10$
11$4$$1.6667$$1001$$-10.1532$$-10.1532$$9/10$
12$4$$1.7500$$1365.1000$$-10.4029$$-10.4029$$8/10$
13$4$$1.2857$$1412$$-10.5321$$-10.5321$$7/10$
14$2$$2.7500$$743.2500$$9.6751e-15$$9.4192e-15$$8/10$
15$3$$1.9000$$3027$$1.3445e-14$$5.6998e-15$$10/10$
16$5$$1.8000$$4999.3$$1.8351e-13$$3.7007e-15$$10/10$
17$7$$1.7500$$8171$$1.7275e-14$$1.3790e-14$$8/10$
18$10$$2.7778$$8895.4$$4.3639e-13$$3.0992e-14$$9/10$
19$20$$2.7143$$18242$$2.2066e-12$$3.0016e-13$$7/10$
20$30$$3.5000$$43232$$6.9372e-12$$1.7361e-12$$6/10$
21$50$$2.5000$$83243$$7.0303e-12$$9.8531e-13$$6/10$
Table 3.  The comparison of the results
No n Our Method Ma et. al [16] El-Gindy et. al [5]
iter-mf.eval-miter-mf.eval-miter-mf.eval-m
1$2$$1.5$$214$$4$$5097$$2$$310$
2$2$$1.13$$290.6$$3$$4012$$2$$778$
3$2$$1.75$$414.3$$3$$2507$$3$$977$
4$2$$1.4$$411$$3$$545$$2$$577$
5$2$$1.5$$234$$3$$518$$2$$279$
6$2$$1.2$$216.5$$1$$595$$2$$265$
7$2$$2.7$$487.9$$3$$8140$$-$$ -$
8$2$$3.4$$813.5$$3$$5280$$3$$635$
9$2$$1$$501$$3$$337$$2$$315$
10$2$$1$$222.3$$3$$1819$$-$$-$
14$2$$2.75$$743.3$$3$$536$$3$$549$
15$3$$1.9$$3027$$1$$6083$$2$$1283$
16$5$$1.8$$4999.3$$1$$7839$$2$$5291$
17$7$$1.75$$8171$$4$$10130$$2$$12793$
18$10$$2.78$$8895.4$$2$$29463$$2$$33810$
19$20$$2.71$$18242$$-$$-$$2$$96223$
20$30$$3.5$$43232$$-$$-$$4$$376885$
21$50$$2.5$$83243$$-$$-$$9$$>10^6$
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No n Our Method Ma et. al [16] El-Gindy et. al [5]
iter-mf.eval-miter-mf.eval-miter-mf.eval-m
1$2$$1.5$$214$$4$$5097$$2$$310$
2$2$$1.13$$290.6$$3$$4012$$2$$778$
3$2$$1.75$$414.3$$3$$2507$$3$$977$
4$2$$1.4$$411$$3$$545$$2$$577$
5$2$$1.5$$234$$3$$518$$2$$279$
6$2$$1.2$$216.5$$1$$595$$2$$265$
7$2$$2.7$$487.9$$3$$8140$$-$$ -$
8$2$$3.4$$813.5$$3$$5280$$3$$635$
9$2$$1$$501$$3$$337$$2$$315$
10$2$$1$$222.3$$3$$1819$$-$$-$
14$2$$2.75$$743.3$$3$$536$$3$$549$
15$3$$1.9$$3027$$1$$6083$$2$$1283$
16$5$$1.8$$4999.3$$1$$7839$$2$$5291$
17$7$$1.75$$8171$$4$$10130$$2$$12793$
18$10$$2.78$$8895.4$$2$$29463$$2$$33810$
19$20$$2.71$$18242$$-$$-$$2$$96223$
20$30$$3.5$$43232$$-$$-$$4$$376885$
21$50$$2.5$$83243$$-$$-$$9$$>10^6$
Table 4.  Numerical Results
$k$$\alpha$$\beta$$x_0$$x_k^*$$f_k^*$
1$0.5$$0.1$(160.0000,280.0000)$(190.2613,277.4205)$$-438.2412$
2$0.5$$0.1$$(190.2613,277.4205)$$(329.0062,186.9678)$$-438.2625$
3$0.5$$0.1$$(329.0062,186.9678)$$(181.6167,187.5836)$$-438.2678$
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$k$$\alpha$$\beta$$x_0$$x_k^*$$f_k^*$
1$0.5$$0.1$(160.0000,280.0000)$(190.2613,277.4205)$$-438.2412$
2$0.5$$0.1$$(190.2613,277.4205)$$(329.0062,186.9678)$$-438.2625$
3$0.5$$0.1$$(329.0062,186.9678)$$(181.6167,187.5836)$$-438.2678$
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