American Institute of Mathematical Sciences

Advanced
2015, 5(3): 237-249. doi: 10.3934/naco.2015.5.237

A quasi-Newton trust region method based on a new fractional model

Honglan Zhu 1, , Qin Ni 1, and  Meilan Zeng 2,

1. 

Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2. 

Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Received  October 2014 Revised  March 2015 Published  August 2015

In this paper, a general fractional model is proposed. Based on the fractional model, a quasi-Newton trust region algorithm is presented for unconstrained optimization. The trust region subproblem is solved in the simplified way. We discussed the choices of the parameters in the fractional model and prove the global convergence of the proposed algorithm. Some primary test results shows the feasibility and validity of the fractional model.
Keywords: fractional model, global convergence., Unconstrained optimization, quasi-Newton method, trust region method.
Mathematics Subject Classification: Primary: 49K10; Secondary: 90C3.
Citation: Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237
References:
[1]

W. C. Davidon, Conic approximations and collinear scalings for optimizers, SIAM Journal on Numerical Analysis, 17 (1980), 268-281. doi: 10.1137/0717023.  Google Scholar

[2]

S. Di and W. Y. Sun, A trust region method for conic model to solve unconstraind optimizaions, Optimization Methods and Software, 6 (1996), 237-263. doi: 10.1080/10556789608805637.  Google Scholar

[3]

D. M. Gay, Computing optimal locally constrained steps, SIAM Journal on Scientific and Statistical Computing, 2 (1981), 186-197. doi: 10.1137/0902016.  Google Scholar

[4]

H. Gourgeon and J. Nocedal, A conic algorithm for optimization, SIAM Journal on Scientific and Statistical Computing, 6 (1985), 253-267. doi: 10.1137/0906019.  Google Scholar

[5]

X. P. Lu, Q. Ni and H. Liu, A dogleg method for solving new trust region subproblems of conic model, ACTA Mathematicae Applicatae Sinica, 30 (2009), 855-871. Google Scholar

[6]

X. P. Lu and Q. Ni, A quasi-newton trust region method with a new conic model for the unconstrained optimization, Applied Mathematics and Computation, 204 (2008), 373-384. doi: 10.1016/j.amc.2008.06.062.  Google Scholar

[7]

J. J. More, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41. doi: 10.1145/355934.355936.  Google Scholar

[8]

Q. Ni, Optimality conditions for trust-region subproblems involving a conic model, SIAM Journal on Optimization, 15 (2005), 826-837. doi: 10.1137/S1052623402418991.  Google Scholar

[9]

Q. Ni, Optimization Method and Program Design, Science Press, Beijing, 2009. Google Scholar

[10]

J. M. Peng and Y. X. Yuan, Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM Journal on Optimization, 7 (1997), 579-594. doi: 10.1137/S1052623494261520.  Google Scholar

[11]

M. J. D. Powell, Convergence properties of a class of minimization algorithms, in Nonlinear Programming 2, (eds. O.L. Mangasarian, R.R. Meyer and S.M. Robinson), Academic Press, New York, (1975), 1-27.  Google Scholar

[12]

M. J. D. Powell and Y. X. Yuan, A trust region algorithm for equality constrained optimization, Mathematical Programming, 49 (1990), 189-211. doi: 10.1007/BF01588787.  Google Scholar

[13]

R. Schnabel, Conic methods for unconstrained minimization and tensor methods for nonlinear equations, In Mathematical Programming: The State of the Art, (eds. A. Bachem, M. Gr\"otschel and B. Korte), Springer-Verlag, Heidelberg, (1982), 417-438.  Google Scholar

[14]

D. C. Sorensen, Newton's method with a model trust region modification, SIAM Journal on Numerical Analysis, 19 (1982), 409-426. doi: 10.1137/0719026.  Google Scholar

[15]

W. Y. Sun and Y. X. Yuan, A conic trust-region method for nonlinearly constrained optimization, Annals of Operations Research, 103 (2001), 175-191. doi: 10.1023/A:1012955122229.  Google Scholar

[16]

F. S. Wang, K. C. Zhang, C. L. Wang and L. Wang, A variant of trust-region methods for unconstrained optimization, Applied Mathematics and Computation, 203 (2008), 297-307. doi: 10.1016/j.amc.2008.04.049.  Google Scholar

[17]

J. Y. Wang and Q. Ni, An algorithm for solving new trust region subproblem with conic model, Science in China, Series A: Mathematics, 51 (2008), 461-473. doi: 10.1007/s11425-007-0149-6.  Google Scholar

[18]

H. P. Wu and Q. Ni, A new trust region algorithm with conic model, Numerical Mathematics: A Journal of Chinese Universities, 30 (2008), 57-67.  Google Scholar

[19]

C. X. Xu and X. Y. Yang, Convergence of conic quasi-Newton trust region methods for unconstrained minimization, Mathematical Application, 11 (1998), 71-76.  Google Scholar

[20]

Y. X. Yuan, A review of trust region algorithms for optimization, ICIAM, 99 (2000), 271-282.  Google Scholar

[21]

L. W. Zhang and Q. Ni, Trust region algorithm of new conic model for nonliearly equality constrained optimization, Journal on Numerical Methods and Computer Applications, 31 (2010), 279-289.  Google Scholar

[22]

X. Zhang, J. Wen and Q. Ni, Subspace trust-region algorithm with conic model for unconstrained optimization, Numerical Algebra, Control and Optimization, 3 (2013), 223-234. doi: 10.3934/naco.2013.3.223.  Google Scholar

[23]

L. J. Zhao and W. Y. Sun, Nonmonotone retrospective conic trust region method for unconstrained optimization, Numerical Algebra, Control and Optimization, 3 (2013), 309-325. doi: 10.3934/naco.2013.3.309.  Google Scholar

[24]

X. Zhao and X. Y. Wang, A nonmonotone self-adaptive search based on trust region algorithm with line the new conic model, Journal of Taiyuan University of Science and Technology, 31 (2010), 68-71. Google Scholar

[25]

M. F. Zhu, Y. Xue and F. S. Zhang, A quasi-Newton type trust region method based on the conic model, Numerical Mathematics A Journal of Chinese Universities, 17 (1995), 36-47.  Google Scholar

show all references

References:
[1]

W. C. Davidon, Conic approximations and collinear scalings for optimizers, SIAM Journal on Numerical Analysis, 17 (1980), 268-281. doi: 10.1137/0717023.  Google Scholar

[2]

S. Di and W. Y. Sun, A trust region method for conic model to solve unconstraind optimizaions, Optimization Methods and Software, 6 (1996), 237-263. doi: 10.1080/10556789608805637.  Google Scholar

[3]

D. M. Gay, Computing optimal locally constrained steps, SIAM Journal on Scientific and Statistical Computing, 2 (1981), 186-197. doi: 10.1137/0902016.  Google Scholar

[4]

H. Gourgeon and J. Nocedal, A conic algorithm for optimization, SIAM Journal on Scientific and Statistical Computing, 6 (1985), 253-267. doi: 10.1137/0906019.  Google Scholar

[5]

X. P. Lu, Q. Ni and H. Liu, A dogleg method for solving new trust region subproblems of conic model, ACTA Mathematicae Applicatae Sinica, 30 (2009), 855-871. Google Scholar

[6]

X. P. Lu and Q. Ni, A quasi-newton trust region method with a new conic model for the unconstrained optimization, Applied Mathematics and Computation, 204 (2008), 373-384. doi: 10.1016/j.amc.2008.06.062.  Google Scholar

[7]

J. J. More, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41. doi: 10.1145/355934.355936.  Google Scholar

[8]

Q. Ni, Optimality conditions for trust-region subproblems involving a conic model, SIAM Journal on Optimization, 15 (2005), 826-837. doi: 10.1137/S1052623402418991.  Google Scholar

[9]

Q. Ni, Optimization Method and Program Design, Science Press, Beijing, 2009. Google Scholar

[10]

J. M. Peng and Y. X. Yuan, Optimality conditions for the minimization of a quadratic with two quadratic constraints, SIAM Journal on Optimization, 7 (1997), 579-594. doi: 10.1137/S1052623494261520.  Google Scholar

[11]

M. J. D. Powell, Convergence properties of a class of minimization algorithms, in Nonlinear Programming 2, (eds. O.L. Mangasarian, R.R. Meyer and S.M. Robinson), Academic Press, New York, (1975), 1-27.  Google Scholar

[12]

M. J. D. Powell and Y. X. Yuan, A trust region algorithm for equality constrained optimization, Mathematical Programming, 49 (1990), 189-211. doi: 10.1007/BF01588787.  Google Scholar

[13]

R. Schnabel, Conic methods for unconstrained minimization and tensor methods for nonlinear equations, In Mathematical Programming: The State of the Art, (eds. A. Bachem, M. Gr\"otschel and B. Korte), Springer-Verlag, Heidelberg, (1982), 417-438.  Google Scholar

[14]

D. C. Sorensen, Newton's method with a model trust region modification, SIAM Journal on Numerical Analysis, 19 (1982), 409-426. doi: 10.1137/0719026.  Google Scholar

[15]

W. Y. Sun and Y. X. Yuan, A conic trust-region method for nonlinearly constrained optimization, Annals of Operations Research, 103 (2001), 175-191. doi: 10.1023/A:1012955122229.  Google Scholar

[16]

F. S. Wang, K. C. Zhang, C. L. Wang and L. Wang, A variant of trust-region methods for unconstrained optimization, Applied Mathematics and Computation, 203 (2008), 297-307. doi: 10.1016/j.amc.2008.04.049.  Google Scholar

[17]

J. Y. Wang and Q. Ni, An algorithm for solving new trust region subproblem with conic model, Science in China, Series A: Mathematics, 51 (2008), 461-473. doi: 10.1007/s11425-007-0149-6.  Google Scholar

[18]

H. P. Wu and Q. Ni, A new trust region algorithm with conic model, Numerical Mathematics: A Journal of Chinese Universities, 30 (2008), 57-67.  Google Scholar

[19]

C. X. Xu and X. Y. Yang, Convergence of conic quasi-Newton trust region methods for unconstrained minimization, Mathematical Application, 11 (1998), 71-76.  Google Scholar

[20]

Y. X. Yuan, A review of trust region algorithms for optimization, ICIAM, 99 (2000), 271-282.  Google Scholar

[21]

L. W. Zhang and Q. Ni, Trust region algorithm of new conic model for nonliearly equality constrained optimization, Journal on Numerical Methods and Computer Applications, 31 (2010), 279-289.  Google Scholar

[22]

X. Zhang, J. Wen and Q. Ni, Subspace trust-region algorithm with conic model for unconstrained optimization, Numerical Algebra, Control and Optimization, 3 (2013), 223-234. doi: 10.3934/naco.2013.3.223.  Google Scholar

[23]

L. J. Zhao and W. Y. Sun, Nonmonotone retrospective conic trust region method for unconstrained optimization, Numerical Algebra, Control and Optimization, 3 (2013), 309-325. doi: 10.3934/naco.2013.3.309.  Google Scholar

[24]

X. Zhao and X. Y. Wang, A nonmonotone self-adaptive search based on trust region algorithm with line the new conic model, Journal of Taiyuan University of Science and Technology, 31 (2010), 68-71. Google Scholar

[25]

M. F. Zhu, Y. Xue and F. S. Zhang, A quasi-Newton type trust region method based on the conic model, Numerical Mathematics A Journal of Chinese Universities, 17 (1995), 36-47.  Google Scholar

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