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

Previous Article
On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems
JIMO Home
This Issue
Next Article
Fuzzy quadratic surface support vector machine based on fisher discriminant analysis

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

Abstract
Related Papers

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 and 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.
[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.

[1]	Jiahui Tang, Yifan Xu, Wei Wang. An approach to solve local and global optimization problems based on exact objective filled penalty functions. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022084
[2]	Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial and Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533
[3]	Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[4]	Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial and Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[5]	Rui L. Fernandes, Yuxuan Zhang. Local and global integrability of Lie brackets. Journal of Geometric Mechanics, 2021, 13 (3) : 355-384. doi: 10.3934/jgm.2021024
[6]	He Huang, Zhen He. A global optimization method for multiple response optimization problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022016
[7]	Deqiang Qu, Youlin Shang, Dan Wu, Guanglei Sun. Filled function method to optimize supply chain transportation costs. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021115
[8]	Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575
[9]	J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
[10]	Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[11]	Rentsen Enkhbat, Evgeniya A. Finkelstein, Anton S. Anikin, Alexandr Yu. Gornov. Global optimization reduction of generalized Malfatti's problem. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 211-221. doi: 10.3934/naco.2017015
[12]	Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial and Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73
[13]	Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070
[14]	Ahmet Sahiner, Nurullah Yilmaz, Gulden Kapusuz. A novel modeling and smoothing technique in global optimization. Journal of Industrial and Management Optimization, 2019, 15 (1) : 113-130. doi: 10.3934/jimo.2018035
[15]	Giancarlo Bigi. Componentwise versus global approaches to nonsmooth multiobjective optimization. Journal of Industrial and Management Optimization, 2005, 1 (1) : 21-32. doi: 10.3934/jimo.2005.1.21
[16]	Chien-Wen Chao, Shu-Cherng Fang, Ching-Jong Liao. A tropical cyclone-based method for global optimization. Journal of Industrial and Management Optimization, 2012, 8 (1) : 103-115. doi: 10.3934/jimo.2012.8.103
[17]	Bun Theang Ong, Masao Fukushima. Global optimization via differential evolution with automatic termination. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 57-67. doi: 10.3934/naco.2012.2.57
[18]	Dmitri E. Kvasov, Yaroslav D. Sergeyev. Univariate geometric Lipschitz global optimization algorithms. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 69-90. doi: 10.3934/naco.2012.2.69
[19]	Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
[20]	T. L. Mason, C. Emelle, J. van Berkel, A. M. Bagirov, F. Kampas, J. D. Pintér. Integrated production system optimization using global optimization techniques. Journal of Industrial and Management Optimization, 2007, 3 (2) : 257-277. doi: 10.3934/jimo.2007.3.257

2020 Impact Factor: 1.801

Tools

Metrics

Other articles
by authors

on AIMS
Liuyang Yuan Zhongping Wan Qiuhua Tang
on Google Scholar
Liuyang Yuan Zhongping Wan Qiuhua Tang

[Back to Top]