A novel modeling and smoothing technique in global optimization

Ahmet Sahiner; Nurullah Yilmaz; Gulden Kapusuz

doi:10.3934/jimo.2018035

Article Contents

2019, Volume 15, Issue 1: 113-130. Doi: 10.3934/jimo.2018035

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A novel modeling and smoothing technique in global optimization

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

* Corresponding author: ahmetsahiner@sdu.edu.tr
* Corresponding author: ahmetsahiner@sdu.edu.tr

Received: March 31, 2017

Revised: December 31, 2017

Early access: April 2018

Published: December 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C26; Secondary: 65D05, 65D17, 97M10.

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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 Name	Dimension $n$	Region	Optimum value
1	Two dimensional function $c=0.05$	$2$	$[-3, 3]^2$	$0$
2	Two dimensional function $c=0.2$	$2$	$[-3, 3]^2$	$0$
3	Two dimensional function $c=0.5$	$2$	$[-3, 3]^2$	$0$
4	3-hump function	$2$	$[-3, 3]^2$	$0$
5	6-hump function	$2$	$[-3, 3]^2$	$-1.0316$
6	Treccani function	$2$	$[-3, 3]^2$	$0$
7	Goldstein-Price function	$2$	$[-3, 3]^2$	$3.0000$
8	Shubert function	$2$	$[-10, 10]^2$	$-186.73091$
9	Rastrigin function	$2$	$[-3, 3]^2$	$-2.0000$
10	Branin 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, 17	Sin-square I function	$2, 3, 5, 7$	$[-10, 10]^n$	$0$
18, 19, 20, 21	Sin-square I function	$10, 20, 30, 50$	$[-10, 10]^n$	$0$

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Table 2. The numerical results of our method

Problem No.	n	iter-m	f.eval-m	f-mean	f-best	SR
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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Table 3. The comparison of the results

No	n	Our Method		Ma et. al [16]		El-Gindy et. al [5]
No	n	iter-m	f.eval-m	iter-m	f.eval-m	iter-m	f.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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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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