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doi: 10.3934/jimo.2017070

## An adaptive trust region algorithm for large-residual nonsmooth least squares problems

 College of Mathematics and Information Science Guangxi University Nanning, Guangxi 530004, China

* Corresponding author

Received  July 2015 Revised  June 2016 Published  September 2017

In this paper, an adaptive trust region algorithm in which the trust region radius converges to zero is presented for solving large-residual nonsmooth least squares problems. This algorithm uses the smoothing technique of the approximation function, and it combines an adaptive trust region radius. Moreover, this algorithm differs from the existing methods for solving nonsmooth equations through use of the approximation function of second-order information, which improves the convergence rate for large-residual nonsmooth least squares problems. Under some suitable conditions, the global and local superlinear convergences of the proposed method are proven. The preliminary numerical results indicate that the proposed algorithm is effective and suitable for solving large-residual nonsmooth least squares problems.

Citation: Zhou Sheng, Gonglin Yuan, Zengru Cui, Xiabin Duan, Xiaoliang Wang. An adaptive trust region algorithm for large-residual nonsmooth least squares problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2017070
##### References:
 [1] M. Al-Baali and R. Fletcher, Variational methods for non-linear least-squares, J. Oper. Res. Soc., 36 (1985), 405-421. [2] X. Chen, On the convergence of Broyden-like methods for nonlinear equations with nondiffentiable terms, Ann. Institut. Statist. Math., 42 (1990), 387-401. doi: 10.1007/BF00050844. [3] X. Chen and L. Qi, A parameterized Newton method and a quasi-Newton method for nonsmooth equations, Comput. Optim. Appl., 3 (1994), 157-179. doi: 10.1007/BF01300972. [4] X. Chen and T. Yamamoto, On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators, Computing, 49 (1992), 87-94. doi: 10.1007/BF02238652. [5] A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA, 2000. [6] J. Fan and J. Pan, An improve trust region algorithm for nonlinear equations, Comput. Optim. Appl., 48 (2011), 59-70. doi: 10.1007/s10589-009-9236-7. [7] J. Fan and Y. Yuan, A new trust region algorithm with trust region radius converging to zero, in Proceedings of the 5th International Conference on Optimization: Techniques and Applications (December 2001, Hongkong) (ed. D. Li), 2001,786-794. [8] L. Hei, A self-adaptive trust region algorithm, J. Comput. Math., 21 (2003), 229-236. [9] K. Levenberg, A method for the solution of certain nonlinear problem in least squares, Quarterly Journal of Mechanics and Applied Mathematics, 2 (1944), 164-168. doi: 10.1090/qam/10666. [10] S. Liu, Z. Wang and C. Liu, On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems, J. Ind. Manag. Optim., 12 (2016), 389-402. doi: 10.3934/jimo.2016.12.389. [11] K. Madsen, An algorithm for minimax solution of overdetermined systems of nonlinear equations, Rep. TP 559, AERE Harwell, England, 1973. [12] B. Martinet, Régularisation d'inéquations variationelles par approxiamations succcessives, Rev. Fr. Inform. Rech. Oper., 4 (1970), 154-158. [13] J. Moré, Recent developments in algorithms and software for trust region methods, in Mathematical Programming: The State of Art (eds. A. Bachem, M. Grötachel and B. Korte), Springer, Berlin, (1983), 258-287. [14] M. J. D. Powell, Convergence properties of a class of minimization algorithms, in Nonlinear Programming (Q. L. Mangasarian, R. R. Meyer and S. M. Robinson), Vol. 2, Academic Press, New York, 1974, 1-27. [15] L. Qi, Trust region algorithms for solving nonsmooth equations, SIAM J. Optimization, 5 (1995), 219-230. doi: 10.1137/0805011. [16] L. Qi, Z. Wei and G. Yuan, An active-set projected trust region algorithm with limited memory BFGS technique for box constrained nonsmooth equations, Optimization, 62 (2013), 857-878. doi: 10.1080/02331934.2011.603321. [17] Z. Sheng, A. Ouyang and L. B. Liu, et al. , A novel parameter estimation method for Muskingum model using new Newton-type trust region algorithm Math. Probl. Eng. (2014), Art. ID 634852, 7 pp. [18] Z. Shi and J. Guo, A new trust region method for unconstrained optimization, J. Comput. and Appl. Math., 213 (2008), 509-520. doi: 10.1016/j.cam.2007.01.027. [19] T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer Anal, 20 (1983), 626-637. doi: 10.1137/0720042. [20] W. Sun, R. J. B. Sampaio and J. Yuan, Quasi-Newton trust region algorithm for non-smooth least squares problems, Appl. Math. Comput., 105 (1999), 183-194. doi: 10.1016/S0096-3003(98)10103-0. [21] G. Yuan, S. Lu and Z. Wei, A new trust-region method with line search for solving symmetric nonlinear equations, Intern. J. Compu. Math., 88 (2011), 2109-2123. doi: 10.1080/00207160.2010.526206. [22] G. Yuan, X. Lu and Z. Wei, BFGS trust-region method for symmetric nonlinear equations, J. Compu. and Appl. Math., 230 (2009), 44-58. doi: 10.1016/j.cam.2008.10.062. [23] G. Yuan, Z. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optim. Theo. Appl., 168 (2016), 129-152. doi: 10.1007/s10957-015-0781-1. [24] G. Yuan, Z. Wei and G. Li, A modified Polak-Ribi're-Polyak conjugate gradient algorithm for nonsmooth convex programs, J. Compu. and Appl. Math., 255 (2014), 86-96. doi: 10.1016/j.cam.2013.04.032. [25] G. Yuan, Z. Wei and X. Lu, A BFGS trust-region method for nonlinear equations, Computing, 92 (2011), 317-333. doi: 10.1007/s00607-011-0146-z. [26] J. Zhang and Y. Wang, A new trust region method for nonlinear equations, Math. Methods Oper. Res., 58 (2003), 283-298. doi: 10.1007/s001860300302. [27] S. Zhou, Y. Li and L. Kong, A smoothing iterative method for quantile regression with nonconvex $l_p$ penalty, J. Ind. Manag. Optim., 12 (2016), 93-112. doi: 10.3934/jimo.2016006.

show all references

##### References:
 [1] M. Al-Baali and R. Fletcher, Variational methods for non-linear least-squares, J. Oper. Res. Soc., 36 (1985), 405-421. [2] X. Chen, On the convergence of Broyden-like methods for nonlinear equations with nondiffentiable terms, Ann. Institut. Statist. Math., 42 (1990), 387-401. doi: 10.1007/BF00050844. [3] X. Chen and L. Qi, A parameterized Newton method and a quasi-Newton method for nonsmooth equations, Comput. Optim. Appl., 3 (1994), 157-179. doi: 10.1007/BF01300972. [4] X. Chen and T. Yamamoto, On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators, Computing, 49 (1992), 87-94. doi: 10.1007/BF02238652. [5] A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA, 2000. [6] J. Fan and J. Pan, An improve trust region algorithm for nonlinear equations, Comput. Optim. Appl., 48 (2011), 59-70. doi: 10.1007/s10589-009-9236-7. [7] J. Fan and Y. Yuan, A new trust region algorithm with trust region radius converging to zero, in Proceedings of the 5th International Conference on Optimization: Techniques and Applications (December 2001, Hongkong) (ed. D. Li), 2001,786-794. [8] L. Hei, A self-adaptive trust region algorithm, J. Comput. Math., 21 (2003), 229-236. [9] K. Levenberg, A method for the solution of certain nonlinear problem in least squares, Quarterly Journal of Mechanics and Applied Mathematics, 2 (1944), 164-168. doi: 10.1090/qam/10666. [10] S. Liu, Z. Wang and C. Liu, On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems, J. Ind. Manag. Optim., 12 (2016), 389-402. doi: 10.3934/jimo.2016.12.389. [11] K. Madsen, An algorithm for minimax solution of overdetermined systems of nonlinear equations, Rep. TP 559, AERE Harwell, England, 1973. [12] B. Martinet, Régularisation d'inéquations variationelles par approxiamations succcessives, Rev. Fr. Inform. Rech. Oper., 4 (1970), 154-158. [13] J. Moré, Recent developments in algorithms and software for trust region methods, in Mathematical Programming: The State of Art (eds. A. Bachem, M. Grötachel and B. Korte), Springer, Berlin, (1983), 258-287. [14] M. J. D. Powell, Convergence properties of a class of minimization algorithms, in Nonlinear Programming (Q. L. Mangasarian, R. R. Meyer and S. M. Robinson), Vol. 2, Academic Press, New York, 1974, 1-27. [15] L. Qi, Trust region algorithms for solving nonsmooth equations, SIAM J. Optimization, 5 (1995), 219-230. doi: 10.1137/0805011. [16] L. Qi, Z. Wei and G. Yuan, An active-set projected trust region algorithm with limited memory BFGS technique for box constrained nonsmooth equations, Optimization, 62 (2013), 857-878. doi: 10.1080/02331934.2011.603321. [17] Z. Sheng, A. Ouyang and L. B. Liu, et al. , A novel parameter estimation method for Muskingum model using new Newton-type trust region algorithm Math. Probl. Eng. (2014), Art. ID 634852, 7 pp. [18] Z. Shi and J. Guo, A new trust region method for unconstrained optimization, J. Comput. and Appl. Math., 213 (2008), 509-520. doi: 10.1016/j.cam.2007.01.027. [19] T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer Anal, 20 (1983), 626-637. doi: 10.1137/0720042. [20] W. Sun, R. J. B. Sampaio and J. Yuan, Quasi-Newton trust region algorithm for non-smooth least squares problems, Appl. Math. Comput., 105 (1999), 183-194. doi: 10.1016/S0096-3003(98)10103-0. [21] G. Yuan, S. Lu and Z. Wei, A new trust-region method with line search for solving symmetric nonlinear equations, Intern. J. Compu. Math., 88 (2011), 2109-2123. doi: 10.1080/00207160.2010.526206. [22] G. Yuan, X. Lu and Z. Wei, BFGS trust-region method for symmetric nonlinear equations, J. Compu. and Appl. Math., 230 (2009), 44-58. doi: 10.1016/j.cam.2008.10.062. [23] G. Yuan, Z. Meng and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optim. Theo. Appl., 168 (2016), 129-152. doi: 10.1007/s10957-015-0781-1. [24] G. Yuan, Z. Wei and G. Li, A modified Polak-Ribi're-Polyak conjugate gradient algorithm for nonsmooth convex programs, J. Compu. and Appl. Math., 255 (2014), 86-96. doi: 10.1016/j.cam.2013.04.032. [25] G. Yuan, Z. Wei and X. Lu, A BFGS trust-region method for nonlinear equations, Computing, 92 (2011), 317-333. doi: 10.1007/s00607-011-0146-z. [26] J. Zhang and Y. Wang, A new trust region method for nonlinear equations, Math. Methods Oper. Res., 58 (2003), 283-298. doi: 10.1007/s001860300302. [27] S. Zhou, Y. Li and L. Kong, A smoothing iterative method for quantile regression with nonconvex $l_p$ penalty, J. Ind. Manag. Optim., 12 (2016), 93-112. doi: 10.3934/jimo.2016006.
The value of $\|F(x_k)\|$ with iteration $k$ for Example 5.1
The value of $\|F(x_k)\|$ with iteration $k$ for Example 5.2
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