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January  2016, 12(1): 375-387. doi: 10.3934/jimo.2016.12.375

A criterion for an approximation global optimal solution based on the filled functions

Liuyang Yuan 1, , Zhongping Wan 2, and  Qiuhua Tang 3,

1. 

College of Science, Wuhan University of Science and Technology, Wuhan, Hubei, 430081, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072
3. 

Industrial Engineering Department, Wuhan University of Science and Technology, Wuhan, Hubei, 430081, China

Received  December 2013 Revised  February 2015 Published  April 2015

In this paper, a new definition of the filled function is given. Based on the new definition, a new class of filled functions is constructed, and the properties of the new filled functions are analysed and discussed. Moreover, according to the new class of filled functions, a criterion is given to decide whether the point we have obtained is an approximate global optimal solution. Finally, a global optimization algorithm based on the new class of filled functions is presented. The implementation of the algorithm on several test problems is reported with numerical results.
Keywords: a criterion, local minimizer, global minimizer., Global optimization, the filled function.
Mathematics Subject Classification: Primary: 90C30; Secondary: 90C3.
Citation: Liuyang Yuan, Zhongping Wan, Qiuhua Tang. A criterion for an approximation global optimal solution based on the filled functions. Journal of Industrial & Management Optimization, 2016, 12 (1) : 375-387. doi: 10.3934/jimo.2016.12.375
References:
[1]

S. H. Chew and Q. Zheng, Integral Global Optimization, Volume 298 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, 1988. doi: 10.1007/978-3-642-46623-6.  Google Scholar

[2]

L. C. W. Dixon, J. Gomulka and S. E. Herson, Reflection on global optimization problems, in Optimization in Action (Dixon, L.C.W. etc.), Academic Press, New York, (1976), 398-435. Google Scholar

[3]

R. P. Ge, A filled function method for finding a global minimizer of a function of several variables, Mathematical Programming, 46 (1990), 191-204. doi: 10.1007/BF01585737.  Google Scholar

[4]

R. P. Ge and Y. F. Qin, A class of filled functions for finding a global minimizer of a function of several variables, Journal of Optimization Theory and Applications, 54 (1987), 241-252. doi: 10.1007/BF00939433.  Google Scholar

[5]

R. P. Ge and Y. F. Qin, The globally convexized filled functions for global optimization, Applied Mathematics and Computation, 35 (1990), 131-158. doi: 10.1016/0096-3003(90)90114-I.  Google Scholar

[6]

R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, $2^{nd}$ edition, Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar

[7]

R. Horst, N. V. Thoai and H. Tuy, Outer approximation by polyhedral convex sets, Operations Research Spektrum, 9 (1987), 153-159. doi: 10.1007/BF01721096.  Google Scholar

[8]

A. Levy and A. Montalvo, The tunneling algorithm for the global minimization of functions, SIAM Journal on Scientific and Statistical Computing, 6 (1986), 15-29. doi: 10.1137/0906002.  Google Scholar

[9]

X. Liu, Finding global minima with a computable filled function, Journal of Global Optimization, 19 (2001), 151-161. doi: 10.1023/A:1008330632677.  Google Scholar

[10]

H. W. Lin, Y. P. Wang, L. Fan and Y. L. Gao, A new discrete filled function method for finding global minimizer of the integer programming, Applied Mathematics and Computation, 219 (2013), 4371-4378. doi: 10.1016/j.amc.2012.10.035.  Google Scholar

[11]

H. W. Lin, Y. L. Gao and Y. P. Wang, A continuously differentiable filled function method for global optimization, Numerical Algorithms, 66 (2014), 511-523. doi: 10.1007/s11075-013-9746-3.  Google Scholar

[12]

R. E. Moore, Enterbal Analysis, Prentice-Hall, NJ: Englewood Cliffs, 1966.  Google Scholar

[13]

P. M. Pardalos, H. E. Romeijn and H. Tuy, Recent development and trends in global optimization, Journal of Computational and Applied Mathematics, 124 (2000), 209-228. doi: 10.1016/S0377-0427(00)00425-8.  Google Scholar

[14]

Z. Wan, L. Y. Yuan and J. W. Chen, A filled function method for nonlinear systems of equalities and inequalities, Computational & Applied Mathematics, 31 (2012), 391-405. doi: 10.1590/S1807-03022012000200010.  Google Scholar

[15]

W. X. Wang, Y. L. Shang, L. S. Zhang and Y. Zhang, Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446. doi: 10.1007/s11590-011-0427-7.  Google Scholar

[16]

F. Wei and Y. P. Wang, A new filled function method with one parameter for global optimization, Mathematical Problems in Engineering, 2013 (2013), 12 pages. Google Scholar

[17]

F. Wei, Y. P. Wang and H. W. Lin, (2014), A new filled function method with two parameters for global optimization, Journal of Optimization Theory and Applications, 163 (2014), 510-527. doi: 10.1007/s10957-013-0515-1.  Google Scholar

[18]

Y. J. Yang and Y. L. Shang, A new filled function method for unconstrained global optimization, Applied Mathematicas Computation, 173 (2006), 501-512. doi: 10.1016/j.amc.2005.04.046.  Google Scholar

[19]

Y. J. Yang, Z. Y. Wu and F. S. Bai, A filled function method for constrained nonlinear integer programming, Journal of Industrial and Management Optimization, 4 (2008), 353-362. doi: 10.3934/jimo.2008.4.353.  Google Scholar

[20]

L. Y. Yuan, Z. Wan, J. J. Zhang and B. Sun, A filled function method for solving nonlinear complementarity problems, Journal of Industrial and Management Optimization, 5 (2009), 911-928. doi: 10.3934/jimo.2009.5.911.  Google Scholar

[21]

L. S. Zhang, C. NG, D. Li and W. Tian, A new filled function method for global optimization, Journal of Global Optimization, 28 (2004), 17-43. doi: 10.1023/B:JOGO.0000006653.60256.f6.  Google Scholar

[22]

Q. Zheng and D. Zhuang, Integral global minimization: Algorithms, implementations and numerical tests, Journal of Global Optimization, 7 (1995), 421-454. doi: 10.1007/BF01099651.  Google Scholar

show all references

References:
[1]

S. H. Chew and Q. Zheng, Integral Global Optimization, Volume 298 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, 1988. doi: 10.1007/978-3-642-46623-6.  Google Scholar

[2]

L. C. W. Dixon, J. Gomulka and S. E. Herson, Reflection on global optimization problems, in Optimization in Action (Dixon, L.C.W. etc.), Academic Press, New York, (1976), 398-435. Google Scholar

[3]

R. P. Ge, A filled function method for finding a global minimizer of a function of several variables, Mathematical Programming, 46 (1990), 191-204. doi: 10.1007/BF01585737.  Google Scholar

[4]

R. P. Ge and Y. F. Qin, A class of filled functions for finding a global minimizer of a function of several variables, Journal of Optimization Theory and Applications, 54 (1987), 241-252. doi: 10.1007/BF00939433.  Google Scholar

[5]

R. P. Ge and Y. F. Qin, The globally convexized filled functions for global optimization, Applied Mathematics and Computation, 35 (1990), 131-158. doi: 10.1016/0096-3003(90)90114-I.  Google Scholar

[6]

R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, $2^{nd}$ edition, Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar

[7]

R. Horst, N. V. Thoai and H. Tuy, Outer approximation by polyhedral convex sets, Operations Research Spektrum, 9 (1987), 153-159. doi: 10.1007/BF01721096.  Google Scholar

[8]

A. Levy and A. Montalvo, The tunneling algorithm for the global minimization of functions, SIAM Journal on Scientific and Statistical Computing, 6 (1986), 15-29. doi: 10.1137/0906002.  Google Scholar

[9]

X. Liu, Finding global minima with a computable filled function, Journal of Global Optimization, 19 (2001), 151-161. doi: 10.1023/A:1008330632677.  Google Scholar

[10]

H. W. Lin, Y. P. Wang, L. Fan and Y. L. Gao, A new discrete filled function method for finding global minimizer of the integer programming, Applied Mathematics and Computation, 219 (2013), 4371-4378. doi: 10.1016/j.amc.2012.10.035.  Google Scholar

[11]

H. W. Lin, Y. L. Gao and Y. P. Wang, A continuously differentiable filled function method for global optimization, Numerical Algorithms, 66 (2014), 511-523. doi: 10.1007/s11075-013-9746-3.  Google Scholar

[12]

R. E. Moore, Enterbal Analysis, Prentice-Hall, NJ: Englewood Cliffs, 1966.  Google Scholar

[13]

P. M. Pardalos, H. E. Romeijn and H. Tuy, Recent development and trends in global optimization, Journal of Computational and Applied Mathematics, 124 (2000), 209-228. doi: 10.1016/S0377-0427(00)00425-8.  Google Scholar

[14]

Z. Wan, L. Y. Yuan and J. W. Chen, A filled function method for nonlinear systems of equalities and inequalities, Computational & Applied Mathematics, 31 (2012), 391-405. doi: 10.1590/S1807-03022012000200010.  Google Scholar

[15]

W. X. Wang, Y. L. Shang, L. S. Zhang and Y. Zhang, Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446. doi: 10.1007/s11590-011-0427-7.  Google Scholar

[16]

F. Wei and Y. P. Wang, A new filled function method with one parameter for global optimization, Mathematical Problems in Engineering, 2013 (2013), 12 pages. Google Scholar

[17]

F. Wei, Y. P. Wang and H. W. Lin, (2014), A new filled function method with two parameters for global optimization, Journal of Optimization Theory and Applications, 163 (2014), 510-527. doi: 10.1007/s10957-013-0515-1.  Google Scholar

[18]

Y. J. Yang and Y. L. Shang, A new filled function method for unconstrained global optimization, Applied Mathematicas Computation, 173 (2006), 501-512. doi: 10.1016/j.amc.2005.04.046.  Google Scholar

[19]

Y. J. Yang, Z. Y. Wu and F. S. Bai, A filled function method for constrained nonlinear integer programming, Journal of Industrial and Management Optimization, 4 (2008), 353-362. doi: 10.3934/jimo.2008.4.353.  Google Scholar

[20]

L. Y. Yuan, Z. Wan, J. J. Zhang and B. Sun, A filled function method for solving nonlinear complementarity problems, Journal of Industrial and Management Optimization, 5 (2009), 911-928. doi: 10.3934/jimo.2009.5.911.  Google Scholar

[21]

L. S. Zhang, C. NG, D. Li and W. Tian, A new filled function method for global optimization, Journal of Global Optimization, 28 (2004), 17-43. doi: 10.1023/B:JOGO.0000006653.60256.f6.  Google Scholar

[22]

Q. Zheng and D. Zhuang, Integral global minimization: Algorithms, implementations and numerical tests, Journal of Global Optimization, 7 (1995), 421-454. doi: 10.1007/BF01099651.  Google Scholar

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