A nonmonotone adaptive trust-region algorithm for symmetric nonlinear equations

ABSTRACT

In this paper, we propose a nonmonotone adap-tive trust-region method for solving symmetric nonlinear equations problems. The convergent result of the presented method will be estab-lished under favorable conditions. Numerical results are reported.

In this paper, we propose a nonmonotone adap-tive trust-region method for solving symmetric nonlinear equations problems. The convergent result of the presented method will be estab-lished under favorable conditions. Numerical results are reported.

Cite this paper

Yuan, G. , Chen, C. and Wei, Z. (2010) A nonmonotone adaptive trust-region algorithm for symmetric nonlinear equations.*Natural Science*, **2**, 373-378. doi: 10.4236/ns.2010.24045.

Yuan, G. , Chen, C. and Wei, Z. (2010) A nonmonotone adaptive trust-region algorithm for symmetric nonlinear equations.

References

[1] Yuan, G.L. and Lu, X.W. (2009) A modified PRP conju-gate gradient method. Annals of Operations Research, 166(1), 73-90.

[2] Yuan, G.L. and Lu, X.W. (2008) A new line search method with trust region for unconstrained optimization. Communications on Applied Nonlinear Analysis, 15(1), 35-49.

[3] Yuan, G.L. and Wei, Z.X. (2009) New line search meth-ods for unconstrained optimization. Journal of the Ko-rean Statistical Society, 38(1), 29-39.

[4] Yuan, G.L. and Wei, Z.X. (2008) Convergence analysis of a modified BFGS method on convex minimizations. Computational Optimization and Applications.

[5] Yuan, G.L. and Wei, Z.X. (2008) The superlinear con-vergence analysis of a nonmonotone BFGS algorithm on convex objective functions. Acta Mathematica Sinica, English Series, 24(1), 35-42.

[6] Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2007) New two-point stepsize gradient methods for solving uncon-strained optimization problems. Natural Science Journal of Xiangtan University, 29(1), 13-15.

[7] Yuan, G.L. and Wei, Z.X. (2004) A new BFGS trust re-gion method. Guangxi Science, 11, 195-196.

[8] Moré, J.J. and Sorensen, D.C. (1983) Computing a trust- region step. SIAM Journal on Scientific and Statistical Computing, 4(3), 553-572.

[9] Flecher, R. (1987) Practical methods of optimization. 2nd Edition, John and Sons, Chichester.

[10] Gay, D.M. (1981) Computing optimal locally constrained steps. SIAM Journal on Scientific and Statistical Com-puting, 2, 186-197.

[11] Powell, M.J.D. (1975) Convergence properties of a class of minimization algorithms. Mangasarian, O.L., Meyer, R.R. and Robinson, S.M., Ed., Nonlinear Programming, Academic Press, New York, 2, 1-27.

[12] Schultz, G.A., Schnabel, R.B. and Bryrd, R.H. (1985) A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM Journal on Numerical Analysis, 22(1), 47-67.

[13] Byrd, R.H., Schnabel, R.B. and Schultz G.A. (1987) A trust-region algorithm for nonlinearly constrained opti-mization. SIAM Journal on Numerical Analysis, 24(5), 1152-1170.

[14] Celis, M.R., Dennis, J.E. and Tapia, R.A. (1985) A trust-region strategy for nonlinear equality constrained optimization, in numerical optimization 1984. Boggs, P.R. Byrd, R.H. and Schnabel, R.B., Ed., SIAM, Philadelphia, 71-82.

[15] Liu, X. and Yuan, Y. (1997) A global convergent, locally superlinearly convergent algorithm for equality con-strained optimization. Research Report, ICM-97-84, Chinese Academy of Sciences, Beijing.

[16] Vardi, A. (1985) A trust-region algorithm for equality constrained minimization: Convergence properties and implementation. SIAM Journal of Numerical Analysis, 22(3), 575-579.

[17] Nocedal, J. and Yuan, Y. (1998) Combining trust region and line search techniques. Advances in Nonlinear Pro-gramming, 153-175.

[18] Sterhaug, T. (1983) The conjugate gradient method and trust regions in large-scale optimization. SIAM Journal Numerical Analysis, 20(3), 626-637.

[19] Yuan, Y. (2000) A review of trust region algorithms for optimization. Proceedings of the 4th International Con-gress on Industrial & Applied Mathematics (ICIAM 99), Edinburgh, 271-282.

[20] Yuan, Y. (2000) On the truncated conjugate gradient method. Mathematical Programming, 87(3), 561-573.

[21] Zhang, X.S., Zhang, J.L. and Liao, L.Z. (2002) An adap-tive trust region method and its convergence. Science in China, 45, 620-631.

[22] Yuan, G.L., Meng, S.D. and Wei, Z.X. (2009) A trust- region-based BFGS method with line search technique for symmetric nonlinear equations. Advances in Opera-tions Research, 2009, 1-20.

[23] Zhang, J.L. and Zhang, X.S. (2003) A nonmonotone adaptive trust region method and its convergence. Com-puters and Mathematics with Applications, 45(10-11), 1469-1477.

[24] Schnabel, R.B. and Eskow, R. (1990) A new modified cholesky factorization. SIAM Journal on Scientific and Statistical Computing, 11(6), 1136-1158.

[25] Griewank, A. (1986) The ‘global’ convergence of Broy-den-like methods with a suitable line search. Journal of the Australian Mathematical Society Series B, 28, 75-92.

[26] Zhu, D.T. (2005) Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations. Applied Mathematics and Computation, 161(3), 875-895.

[27] Fan, J.Y. (2003) A modified Levenberg-Marquardt algo-rithm for singular system of nonlinear equations. Journal of Computational Mathematics, 21, 625-636.

[28] Yuan, Y. (1998) Trust region algorithm for nonlinear equations. Information, 1, 7-21.

[29] Yuan, G.L., Wei, Z.X. and Lu, X.W. (2009) A non-monotone trust region method for solving symmetric nonlinear equations. Chinese Quarterly Journal of Mathe- matics, 24, 574-584.

[30] Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2007) A modi-fied trust region method with global convergence for symmetric nonlinear equations. Mathematica Numerica Sinica, 11(3), 225-234.

[31] Li, D. and Fukushima, M. (1999) A global and superlin-ear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM Journal on Nu-merical Analysis, 37(1), 152-172.

[32] Wei, Z.X., Yuan, G.L. and Lian, Z.G. (2004) An ap-proximate Gauss-Newton-based BFGS method for solv-ing symmetric nonlinear equations. Guangxi Sciences, 11(2), 91-99.

[33] Yuan, G.L. and Li, X.R. (2004) An approximate Gauss- Newton-based BFGS method with descent directions for solving symmetric nonlinear equations. OR Transactions, 8(4), 10-26.

[34] Yuan, G.L. and Li, X.R. (2010) A rank-one fitting method for solving symmetric nonlinear equations. Journal of Applied Functional Analysis, 5(4), 389-407.

[35] Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2009) BFGS trust-region method for symmetric nonlinear equations. Journal of Computational and Applied Mathematics, 230(1), 44-58.

[36] Yuan, G.L., Wei, Z.X. and Lu, X.W. (2006) A modified Gauss-Newton-based BFGS method for symmetric non- linear equations. Guangxi Science, 13(4), 288-292.

[37] Yuan, G.L., Wang, Z.X. and Wei, Z.X. (2009) A rank-one fitting method with descent direction for solving sym-metric nonlinear equations. International Journal of Communications, Network and System Sciences, 2(6), 555-561.

[38] Yuan, G.L. and Lu, X.W. (2008) A new backtracking inexact BFGS method for symmetric nonlinear equations. Computer and Mathematics with Application, 55(1), 116- 129.

[39] Ortega, J.M. and Rheinboldt, W.C. (1970) Iterative solu-tion of nonlinear equations in several variables. Aca-demic Press, New York.

[1] Yuan, G.L. and Lu, X.W. (2009) A modified PRP conju-gate gradient method. Annals of Operations Research, 166(1), 73-90.

[2] Yuan, G.L. and Lu, X.W. (2008) A new line search method with trust region for unconstrained optimization. Communications on Applied Nonlinear Analysis, 15(1), 35-49.

[3] Yuan, G.L. and Wei, Z.X. (2009) New line search meth-ods for unconstrained optimization. Journal of the Ko-rean Statistical Society, 38(1), 29-39.

[4] Yuan, G.L. and Wei, Z.X. (2008) Convergence analysis of a modified BFGS method on convex minimizations. Computational Optimization and Applications.

[5] Yuan, G.L. and Wei, Z.X. (2008) The superlinear con-vergence analysis of a nonmonotone BFGS algorithm on convex objective functions. Acta Mathematica Sinica, English Series, 24(1), 35-42.

[6] Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2007) New two-point stepsize gradient methods for solving uncon-strained optimization problems. Natural Science Journal of Xiangtan University, 29(1), 13-15.

[7] Yuan, G.L. and Wei, Z.X. (2004) A new BFGS trust re-gion method. Guangxi Science, 11, 195-196.

[8] Moré, J.J. and Sorensen, D.C. (1983) Computing a trust- region step. SIAM Journal on Scientific and Statistical Computing, 4(3), 553-572.

[9] Flecher, R. (1987) Practical methods of optimization. 2nd Edition, John and Sons, Chichester.

[10] Gay, D.M. (1981) Computing optimal locally constrained steps. SIAM Journal on Scientific and Statistical Com-puting, 2, 186-197.

[11] Powell, M.J.D. (1975) Convergence properties of a class of minimization algorithms. Mangasarian, O.L., Meyer, R.R. and Robinson, S.M., Ed., Nonlinear Programming, Academic Press, New York, 2, 1-27.

[12] Schultz, G.A., Schnabel, R.B. and Bryrd, R.H. (1985) A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM Journal on Numerical Analysis, 22(1), 47-67.

[13] Byrd, R.H., Schnabel, R.B. and Schultz G.A. (1987) A trust-region algorithm for nonlinearly constrained opti-mization. SIAM Journal on Numerical Analysis, 24(5), 1152-1170.

[14] Celis, M.R., Dennis, J.E. and Tapia, R.A. (1985) A trust-region strategy for nonlinear equality constrained optimization, in numerical optimization 1984. Boggs, P.R. Byrd, R.H. and Schnabel, R.B., Ed., SIAM, Philadelphia, 71-82.

[15] Liu, X. and Yuan, Y. (1997) A global convergent, locally superlinearly convergent algorithm for equality con-strained optimization. Research Report, ICM-97-84, Chinese Academy of Sciences, Beijing.

[16] Vardi, A. (1985) A trust-region algorithm for equality constrained minimization: Convergence properties and implementation. SIAM Journal of Numerical Analysis, 22(3), 575-579.

[17] Nocedal, J. and Yuan, Y. (1998) Combining trust region and line search techniques. Advances in Nonlinear Pro-gramming, 153-175.

[18] Sterhaug, T. (1983) The conjugate gradient method and trust regions in large-scale optimization. SIAM Journal Numerical Analysis, 20(3), 626-637.

[19] Yuan, Y. (2000) A review of trust region algorithms for optimization. Proceedings of the 4th International Con-gress on Industrial & Applied Mathematics (ICIAM 99), Edinburgh, 271-282.

[20] Yuan, Y. (2000) On the truncated conjugate gradient method. Mathematical Programming, 87(3), 561-573.

[21] Zhang, X.S., Zhang, J.L. and Liao, L.Z. (2002) An adap-tive trust region method and its convergence. Science in China, 45, 620-631.

[22] Yuan, G.L., Meng, S.D. and Wei, Z.X. (2009) A trust- region-based BFGS method with line search technique for symmetric nonlinear equations. Advances in Opera-tions Research, 2009, 1-20.

[23] Zhang, J.L. and Zhang, X.S. (2003) A nonmonotone adaptive trust region method and its convergence. Com-puters and Mathematics with Applications, 45(10-11), 1469-1477.

[24] Schnabel, R.B. and Eskow, R. (1990) A new modified cholesky factorization. SIAM Journal on Scientific and Statistical Computing, 11(6), 1136-1158.

[25] Griewank, A. (1986) The ‘global’ convergence of Broy-den-like methods with a suitable line search. Journal of the Australian Mathematical Society Series B, 28, 75-92.

[26] Zhu, D.T. (2005) Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations. Applied Mathematics and Computation, 161(3), 875-895.

[27] Fan, J.Y. (2003) A modified Levenberg-Marquardt algo-rithm for singular system of nonlinear equations. Journal of Computational Mathematics, 21, 625-636.

[28] Yuan, Y. (1998) Trust region algorithm for nonlinear equations. Information, 1, 7-21.

[29] Yuan, G.L., Wei, Z.X. and Lu, X.W. (2009) A non-monotone trust region method for solving symmetric nonlinear equations. Chinese Quarterly Journal of Mathe- matics, 24, 574-584.

[30] Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2007) A modi-fied trust region method with global convergence for symmetric nonlinear equations. Mathematica Numerica Sinica, 11(3), 225-234.

[31] Li, D. and Fukushima, M. (1999) A global and superlin-ear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM Journal on Nu-merical Analysis, 37(1), 152-172.

[32] Wei, Z.X., Yuan, G.L. and Lian, Z.G. (2004) An ap-proximate Gauss-Newton-based BFGS method for solv-ing symmetric nonlinear equations. Guangxi Sciences, 11(2), 91-99.

[33] Yuan, G.L. and Li, X.R. (2004) An approximate Gauss- Newton-based BFGS method with descent directions for solving symmetric nonlinear equations. OR Transactions, 8(4), 10-26.

[34] Yuan, G.L. and Li, X.R. (2010) A rank-one fitting method for solving symmetric nonlinear equations. Journal of Applied Functional Analysis, 5(4), 389-407.

[35] Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2009) BFGS trust-region method for symmetric nonlinear equations. Journal of Computational and Applied Mathematics, 230(1), 44-58.

[36] Yuan, G.L., Wei, Z.X. and Lu, X.W. (2006) A modified Gauss-Newton-based BFGS method for symmetric non- linear equations. Guangxi Science, 13(4), 288-292.

[37] Yuan, G.L., Wang, Z.X. and Wei, Z.X. (2009) A rank-one fitting method with descent direction for solving sym-metric nonlinear equations. International Journal of Communications, Network and System Sciences, 2(6), 555-561.

[38] Yuan, G.L. and Lu, X.W. (2008) A new backtracking inexact BFGS method for symmetric nonlinear equations. Computer and Mathematics with Application, 55(1), 116- 129.

[39] Ortega, J.M. and Rheinboldt, W.C. (1970) Iterative solu-tion of nonlinear equations in several variables. Aca-demic Press, New York.