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

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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

Abstract
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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 and 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.
[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.
[3]	D. M. Gay, Computing optimal locally constrained steps, SIAM Journal on Scientific and Statistical Computing, 2 (1981), 186-197. doi: 10.1137/0902016.
[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.
[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.
[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.
[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.
[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.
[9]	Q. Ni, Optimization Method and Program Design, Science Press, Beijing, 2009.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	Y. X. Yuan, A review of trust region algorithms for optimization, ICIAM, 99 (2000), 271-282.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[3]	D. M. Gay, Computing optimal locally constrained steps, SIAM Journal on Scientific and Statistical Computing, 2 (1981), 186-197. doi: 10.1137/0902016.
[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.
[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.
[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.
[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.
[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.
[9]	Q. Ni, Optimization Method and Program Design, Science Press, Beijing, 2009.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	Y. X. Yuan, A review of trust region algorithms for optimization, ICIAM, 99 (2000), 271-282.
[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.
[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.
[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.
[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.
[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.

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