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October  2013, 9(4): 919-944. doi: 10.3934/jimo.2013.9.919

A non-monotone retrospective trust-region method for unconstrained optimization

Jun Chen 1, , Wenyu Sun 1, and  Zhenghao Yang 2,

1. 

School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210046, China
2. 

Department of Applied Mathematics, Nanjing Agricultural University, Nanjing 210095, China

Received  January 2012 Revised  April 2013 Published  August 2013

In this paper, a new non-monotone trust-region algorithm is proposed for solving unconstrained nonlinear optimization problems. We modify the retrospective ratio which is introduced by Bastin et al. [Math. Program., Ser. A (2010) 123: 395-418] to form a convex combination ratio for updating the trust-region radius. Then we combine the non-monotone technique with this new framework of trust-region algorithm. The new algorithm is shown to be globally convergent to a first-order critical point. Numerical experiments on CUTEr problems indicate that it is competitive with both the original retrospective trust-region algorithm and the classical trust-region algorithms.
Keywords: convergence analysis, CUTEr., retrospective trust-region method, Unconstrained optimization, non-monotone technique.
Mathematics Subject Classification: 90C30, 90C26, 65K0.
Citation: Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919
References:
[1]

F. Bastin, V. Malmedy, M. Mouffe, Ph. L. Toint and D. Tomanos, A retrospective trust-region method for unconstrained optimization, Mathematical Programming, 123 (2010), 395-418. doi: 10.1007/s10107-008-0258-1.  Google Scholar

[2]

I. Bongartz, A. R. Conn, N. I. M. Gould and Ph. L. Toint, CUTE: Constrained and Unconstrained Testing Environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.  Google Scholar

[3]

R. M. Chamberlain, M. J. D. Powell, C. Lemaréchal and H. C. Pedersen, The watchdog technique for forcing convergence in algorithms for constrained optimization, Mathematical Programming Studies, 16 (1982), 1-17.  Google Scholar

[4]

J. Chen, W. Y. Sun and R. J. B. de Sampaio, Numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters, Computers and Mathematics with Applications, 56 (2008), 2932-2940. doi: 10.1016/j.camwa.2008.05.010.  Google Scholar

[5]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,'' MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719857.  Google Scholar

[6]

N. Y. Deng, Y. Xiao and F. J. Zhou, Nonmonotonic trust region algorithm, Journal of Optimization Theory and Applications, 76 (1993), 259-285. doi: 10.1007/BF00939608.  Google Scholar

[7]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, Series A, 91 (2002), 201-213. doi: 10.1007/s101070100263.  Google Scholar

[8]

J. H. Fu and W. Y. Sun, Nonmonotone adaptive trust-region method for unconstrained optimization problems, Applied Mathematics and Computation, 163 (2005), 489-504. doi: 10.1016/j.amc.2004.02.011.  Google Scholar

[9]

N. I. M. Gould, D. Orban and Ph. L. Toint, CUTEr and SifDec: A Constrained and Unconstrained Testing Environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 373-394. doi: 10.1145/962437.962438.  Google Scholar

[10]

L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for newton's method, SIAM Journal on Numerical Analysis, 23 (1986), 707-716. doi: 10.1137/0723046.  Google Scholar

[11]

J. T. Mo, C. Y. Liu and S. C. Yan, A nonmonotone trust region method based on nonincreasing technique of weighted average of the successive function values, Journal of Computational and Applied Mathematics, 209 (2007), 97-108. doi: 10.1016/j.cam.2006.10.070.  Google Scholar

[12]

J. J. Moré, Recent developments in algorithms and software for trust region methods, in "Mathematical Programming: The State of the Art" (eds. A. Bachem, M. Grötschel and B. Korte), Springer, Berlin, (1983), 258-287.  Google Scholar

[13]

J. J. Moré and D. C. Sorensen, Computing a trust region step, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 553-572. doi: 10.1137/0904038.  Google Scholar

[14]

J. Nocedal and S. J. Wright, "Numerical Optimization,'' $2^{nd}$ edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[15]

M. J. D. Powell, A hybrid method for nonlinear equations, in "Numerical Methods for Nonlinear Algebraic Equations" (ed. P. Rabinowitz), Gordon and Breach, London, (1970), 87-114.  Google Scholar

[16]

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, London, (1974), 1-27.  Google Scholar

[17]

G. A. Shultz, R. B. Schnabel and R. H. Byrd, A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties, SIAM Journal on Numerical Analysis, 22 (1985), 47-67. doi: 10.1137/0722003.  Google Scholar

[18]

W. Y. Sun, Nonmonotone trust region method for solving optimization problems, Applied Mathematics and Computation, 156 (2004), 159-174. doi: 10.1016/j.amc.2003.07.008.  Google Scholar

[19]

W. Y. Sun, J. Y. Han and J. Sun, Global convergence of nonmonotone descent methods for unconstrained optimization problems, Journal of Computational and Applied Mathematics, 146 (2002), 89-98. doi: 10.1016/S0377-0427(02)00420-X.  Google Scholar

[20]

W. Y. Sun and Y. X. Yuan, "Optimization Theory and Methods. Nonlinear Programming,'' Springer Optimization and Its Applications, Vol. 1, Springer, New York, 2006.  Google Scholar

[21]

W. Y. Sun and Q. Y. Zhou, An unconstrained optimization method using nonmonotone second order Goldstein's line search, Science in China Series A: Mathematics, 50 (2007), 1389-1400. doi: 10.1007/s11425-007-0072-x.  Google Scholar

[22]

Ph. L. Toint, An assessment of nonmonotone linesearch techniques for unconstrained optimization, SIAM Journal on Scientific Computing, 17 (1996), 725-739. doi: 10.1137/S106482759427021X.  Google Scholar

[23]

Ph. L. Toint, A non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints, Mathematical Programming, 77 (1997), 69-94. doi: 10.1007/BF02614518.  Google Scholar

[24]

Y. X. Yuan, On the convergence of trust region algorithms, (in Chinese) Mathematica Numerica Sinica, 16 (1994), 333-346.  Google Scholar

[25]

Y. X. Yuan and W. Y. Sun, "Optimization Theory and Methods,'' (in Chinese) Science Press, Beijing, 1997. Google Scholar

[26]

H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM Journal on Optimization, 14 (2004), 1043-1056. doi: 10.1137/S1052623403428208.  Google Scholar

[27]

D. T. Zhu, A nonmonotone trust region technique for unconstrained optimization problems, Systems Science and Mathematical Sciences, 11 (1998), 375-382.  Google Scholar

show all references

References:
[1]

F. Bastin, V. Malmedy, M. Mouffe, Ph. L. Toint and D. Tomanos, A retrospective trust-region method for unconstrained optimization, Mathematical Programming, 123 (2010), 395-418. doi: 10.1007/s10107-008-0258-1.  Google Scholar

[2]

I. Bongartz, A. R. Conn, N. I. M. Gould and Ph. L. Toint, CUTE: Constrained and Unconstrained Testing Environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.  Google Scholar

[3]

R. M. Chamberlain, M. J. D. Powell, C. Lemaréchal and H. C. Pedersen, The watchdog technique for forcing convergence in algorithms for constrained optimization, Mathematical Programming Studies, 16 (1982), 1-17.  Google Scholar

[4]

J. Chen, W. Y. Sun and R. J. B. de Sampaio, Numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters, Computers and Mathematics with Applications, 56 (2008), 2932-2940. doi: 10.1016/j.camwa.2008.05.010.  Google Scholar

[5]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,'' MPS/SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719857.  Google Scholar

[6]

N. Y. Deng, Y. Xiao and F. J. Zhou, Nonmonotonic trust region algorithm, Journal of Optimization Theory and Applications, 76 (1993), 259-285. doi: 10.1007/BF00939608.  Google Scholar

[7]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, Series A, 91 (2002), 201-213. doi: 10.1007/s101070100263.  Google Scholar

[8]

J. H. Fu and W. Y. Sun, Nonmonotone adaptive trust-region method for unconstrained optimization problems, Applied Mathematics and Computation, 163 (2005), 489-504. doi: 10.1016/j.amc.2004.02.011.  Google Scholar

[9]

N. I. M. Gould, D. Orban and Ph. L. Toint, CUTEr and SifDec: A Constrained and Unconstrained Testing Environment, revisited, ACM Transactions on Mathematical Software, 29 (2003), 373-394. doi: 10.1145/962437.962438.  Google Scholar

[10]

L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for newton's method, SIAM Journal on Numerical Analysis, 23 (1986), 707-716. doi: 10.1137/0723046.  Google Scholar

[11]

J. T. Mo, C. Y. Liu and S. C. Yan, A nonmonotone trust region method based on nonincreasing technique of weighted average of the successive function values, Journal of Computational and Applied Mathematics, 209 (2007), 97-108. doi: 10.1016/j.cam.2006.10.070.  Google Scholar

[12]

J. J. Moré, Recent developments in algorithms and software for trust region methods, in "Mathematical Programming: The State of the Art" (eds. A. Bachem, M. Grötschel and B. Korte), Springer, Berlin, (1983), 258-287.  Google Scholar

[13]

J. J. Moré and D. C. Sorensen, Computing a trust region step, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 553-572. doi: 10.1137/0904038.  Google Scholar

[14]

J. Nocedal and S. J. Wright, "Numerical Optimization,'' $2^{nd}$ edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[15]

M. J. D. Powell, A hybrid method for nonlinear equations, in "Numerical Methods for Nonlinear Algebraic Equations" (ed. P. Rabinowitz), Gordon and Breach, London, (1970), 87-114.  Google Scholar

[16]

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, London, (1974), 1-27.  Google Scholar

[17]

G. A. Shultz, R. B. Schnabel and R. H. Byrd, A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties, SIAM Journal on Numerical Analysis, 22 (1985), 47-67. doi: 10.1137/0722003.  Google Scholar

[18]

W. Y. Sun, Nonmonotone trust region method for solving optimization problems, Applied Mathematics and Computation, 156 (2004), 159-174. doi: 10.1016/j.amc.2003.07.008.  Google Scholar

[19]

W. Y. Sun, J. Y. Han and J. Sun, Global convergence of nonmonotone descent methods for unconstrained optimization problems, Journal of Computational and Applied Mathematics, 146 (2002), 89-98. doi: 10.1016/S0377-0427(02)00420-X.  Google Scholar

[20]

W. Y. Sun and Y. X. Yuan, "Optimization Theory and Methods. Nonlinear Programming,'' Springer Optimization and Its Applications, Vol. 1, Springer, New York, 2006.  Google Scholar

[21]

W. Y. Sun and Q. Y. Zhou, An unconstrained optimization method using nonmonotone second order Goldstein's line search, Science in China Series A: Mathematics, 50 (2007), 1389-1400. doi: 10.1007/s11425-007-0072-x.  Google Scholar

[22]

Ph. L. Toint, An assessment of nonmonotone linesearch techniques for unconstrained optimization, SIAM Journal on Scientific Computing, 17 (1996), 725-739. doi: 10.1137/S106482759427021X.  Google Scholar

[23]

Ph. L. Toint, A non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints, Mathematical Programming, 77 (1997), 69-94. doi: 10.1007/BF02614518.  Google Scholar

[24]

Y. X. Yuan, On the convergence of trust region algorithms, (in Chinese) Mathematica Numerica Sinica, 16 (1994), 333-346.  Google Scholar

[25]

Y. X. Yuan and W. Y. Sun, "Optimization Theory and Methods,'' (in Chinese) Science Press, Beijing, 1997. Google Scholar

[26]

H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM Journal on Optimization, 14 (2004), 1043-1056. doi: 10.1137/S1052623403428208.  Google Scholar

[27]

D. T. Zhu, A nonmonotone trust region technique for unconstrained optimization problems, Systems Science and Mathematical Sciences, 11 (1998), 375-382.  Google Scholar

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