Article Contents
Article Contents

# A novel modeling and smoothing technique in global optimization

• 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.

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

 Citation:

• Figure 1.  The subregions of $\Omega = [0,360]\times[0,360]$

Figure 2.  Constructed Bezier surfaces on the subregions $-438$ was taken as zero to remove the complexity

Figure 3.  The graph of the function $\tilde{f}(x, y, \varepsilon, \delta)$ which is constructed by blending Bezier surfaces

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 $10, 20, 30, 50$ $[-10, 10]^n$ $0$

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 $50$ $2.5000$ $83243$ $7.0303e-12$ $9.8531e-13$ $6/10$

Table 3.  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 $50$ $2.5$ $83243$ $-$ $-$ $9$ $>10^6$

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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