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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 and 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.

[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.

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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