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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, $2^{nd}$ edition, Kluwer Academic Publishers, Dordrecht, 2001. |
[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. |
[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. |
[9] |
X. Liu, Finding global minima with a computable filled function, Journal of Global Optimization, 19 (2001), 151-161.
doi: 10.1023/A:1008330632677. |
[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. |
[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. |
[12] |
R. E. Moore, Enterbal Analysis, Prentice-Hall, NJ: Englewood Cliffs, 1966. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
R. Horst, P. M. Pardalos and N. V. Thoai, Introduction to Global Optimization, $2^{nd}$ edition, Kluwer Academic Publishers, Dordrecht, 2001. |
[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. |
[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. |
[9] |
X. Liu, Finding global minima with a computable filled function, Journal of Global Optimization, 19 (2001), 151-161.
doi: 10.1023/A:1008330632677. |
[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. |
[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. |
[12] |
R. E. Moore, Enterbal Analysis, Prentice-Hall, NJ: Englewood Cliffs, 1966. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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