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

A novel modeling and smoothing technique in global optimization

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

* Corresponding author: ahmetsahiner@sdu.edu.tr

Received  April 2017 Revised  January 2018 Published  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.

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
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References:
The subregions of $\Omega = [0,360]\times[0,360]$
Constructed Bezier surfaces on the subregions $-438$ was taken as zero to remove the complexity
The graph of the function $\tilde{f}(x, y, \varepsilon, \delta)$ which is constructed by blending Bezier surfaces
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 $10, 20, 30, 50$ $[-10, 10]^n$ $0$
 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 $10, 20, 30, 50$ $[-10, 10]^n$ $0$
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 $50$ $2.5000$ $83243$ $7.0303e-12$ $9.8531e-13$ $6/10$
 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 $50$ $2.5000$ $83243$ $7.0303e-12$ $9.8531e-13$ $6/10$
The comparison of the results
 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 $2$ $1.5$ $214$ $4$ $5097$ $2$ $310$ 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 $50$ $2.5$ $83243$ $-$ $-$ $9$ $>10^6$
 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 $2$ $1.5$ $214$ $4$ $5097$ $2$ $310$ 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 $50$ $2.5$ $83243$ $-$ $-$ $9$ $>10^6$
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 $0.5$ $0.1$ $(329.0062,186.9678)$ $(181.6167,187.5836)$ $-438.2678$
 $k$ $\alpha$ $\beta$ $x_0$ $x_k^*$ $f_k^*$ 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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