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

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  • 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.

    Mathematics Subject Classification: 90C26.


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  • Figure 1.  The value of $\|F(x_k)\|$ with iteration $k$ for Example 5.1

    Figure 2.  The value of $\|F(x_k)\|$ with iteration $k$ for Example 5.2

  •   M. Al-Baali  and  R. Fletcher , Variational methods for non-linear least-squares, J. Oper. Res. Soc., 36 (1985) , 405-421. 
      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.
      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.
      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.
      A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719857.
      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.
      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. doi: 10.4208/jcm.1601-m2015-0399.
      L. Hei , A self-adaptive trust region algorithm, J. Comput. Math., 21 (2003) , 229-236. 
      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.
      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.
      K. Madsen, An algorithm for minimax solution of overdetermined systems of nonlinear equations, Rep. TP 559, AERE Harwell, England, 1973.
      B. Martinet , Régularisation d'inéquations variationelles par approxiamations succcessives, Rev. Fr. Inform. Rech. Oper., 4 (1970) , 154-158. 
      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. 
      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.
      L. Qi , Trust region algorithms for solving nonsmooth equations, SIAM J. Optimization, 5 (1995) , 219-230.  doi: 10.1137/0805011.
      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.
      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. doi: 10.1155/2014/634852.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
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