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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

 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.
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, (1976), 398.   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.  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.  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.  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, (2001).   Google Scholar [7] R. Horst, N. V. Thoai and H. Tuy, Outer approximation by polyhedral convex sets,, Operations Research Spektrum, 9 (1987), 153.  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.  doi: 10.1137/0906002.  Google Scholar [9] X. Liu, Finding global minima with a computable filled function,, Journal of Global Optimization, 19 (2001), 151.  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.  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.  doi: 10.1007/s11075-013-9746-3.  Google Scholar [12] R. E. Moore, Enterbal Analysis,, Prentice-Hall, (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.  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.  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.  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).   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.  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.  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.  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.  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.  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.  doi: 10.1007/BF01099651.  Google Scholar

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##### 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, (1976), 398.   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.  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.  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.  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, (2001).   Google Scholar [7] R. Horst, N. V. Thoai and H. Tuy, Outer approximation by polyhedral convex sets,, Operations Research Spektrum, 9 (1987), 153.  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.  doi: 10.1137/0906002.  Google Scholar [9] X. Liu, Finding global minima with a computable filled function,, Journal of Global Optimization, 19 (2001), 151.  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.  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.  doi: 10.1007/s11075-013-9746-3.  Google Scholar [12] R. E. Moore, Enterbal Analysis,, Prentice-Hall, (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.  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.  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.  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).   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.  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.  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.  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.  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.  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.  doi: 10.1007/BF01099651.  Google Scholar
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