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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

 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.
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.  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.  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.   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.  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, (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.  doi: 10.1007/BF00939608.  Google Scholar [7] E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Mathematical Programming, 91 (2002), 201.  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.  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.  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.  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.  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, (1983), 258.   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.  doi: 10.1137/0904038.  Google Scholar [14] J. Nocedal and S. J. Wright, "Numerical Optimization,'', $2^{nd}$ edition, (2006).   Google Scholar [15] M. J. D. Powell, A hybrid method for nonlinear equations,, in, (1970), 87.   Google Scholar [16] M. J. D. Powell, Convergence properties of a class of minimization algorithms,, in, (1974), 1.   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.  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.  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.  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, (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.  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.  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.  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.   Google Scholar [25] Y. X. Yuan and W. Y. Sun, "Optimization Theory and Methods,'', (in Chinese) Science Press, (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.  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.   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.  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.  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.   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.  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, (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.  doi: 10.1007/BF00939608.  Google Scholar [7] E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Mathematical Programming, 91 (2002), 201.  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.  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.  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.  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.  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, (1983), 258.   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.  doi: 10.1137/0904038.  Google Scholar [14] J. Nocedal and S. J. Wright, "Numerical Optimization,'', $2^{nd}$ edition, (2006).   Google Scholar [15] M. J. D. Powell, A hybrid method for nonlinear equations,, in, (1970), 87.   Google Scholar [16] M. J. D. Powell, Convergence properties of a class of minimization algorithms,, in, (1974), 1.   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.  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.  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.  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, (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.  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.  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.  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.   Google Scholar [25] Y. X. Yuan and W. Y. Sun, "Optimization Theory and Methods,'', (in Chinese) Science Press, (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.  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.   Google Scholar
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