# American Institute of Mathematical Sciences

• Previous Article
Mechanism design in project procurement auctions with cost uncertainty and failure risk
• JIMO Home
• This Issue
• Next Article
Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank
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
##### References:

show all references

##### 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 $0.5$ $0.1$ $(329.0062,186.9678)$ $(181.6167,187.5836)$ $-438.2678$
 [1] Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010 [2] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350 [3] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [4] Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115 [5] Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026 [6] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031 [7] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [8] Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082 [9] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [10] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [11] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [12] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [13] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHum approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [14] M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 [15] Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 [16] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [17] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [18] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

2019 Impact Factor: 1.366