zbMATH — the first resource for mathematics

Nonmonotone filter DQMM method for the system of nonlinear equations. (English) Zbl 1223.65036
A nonmonotone filter diagonalized quasi-Newton multiplier (DQNM) method is proposed for solving a system of nonlinear equations. The system of nonlinear equations is transformed into a constrained nonlinear programming problem that is then solved by a nonmonotone DQNM method. A nonlinear criterion is used to speed up the rate of convergence in some ill-conditioned cases. Under reasonable conditions, global convergence is proven. Numerical experiments show the effectiveness of the proposed algorithm.

65H10 Numerical computation of solutions to systems of equations
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C53 Methods of quasi-Newton type
Full Text: DOI
[1] R.H. Byrd, M. Marazzi, J. Nocedal, On the convergence of Newton iterations to non-stationary points. Report OTC 2001/01, Optimization Technology Center, Northwestern University, Evanston, Argonne, IL60208, USA, 2001. · Zbl 1072.90038
[2] Fletcher, R.; Leyffer, S., Nonlinear programming without a penalty function, Math. program., 91, 239-269, (2002) · Zbl 1049.90088
[3] Fletcher, R.; Leyffer, S.; Toint, P.L., On the global convergence of a filter-SQP algorithm, SIAM J. optim., 13, 44-59, (2002) · Zbl 1029.65063
[4] Fletcher, R.; Gould, N.I.M.; Leyffer, S.; Toint, P.L.; Wächter, A., Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming, SIAM J. optim., 13, 635-659, (2002) · Zbl 1038.90076
[5] Grippo, L.; Lampariello, F.; Ludidi, S., A nonmonotone line search technique for newtons method, SIAM J. numer. anal., 23, 707-716, (1986) · Zbl 0616.65067
[6] C. Gu, D. Zhu, A secant algorithm with line search filter method for nonlinear optimization, Technical Report, Department of Mathematics, Shanghai Normal University, Shanghai, China, 2007.
[7] Gu, C.; Zhu, D., A filter interior-point algorithm with projected Hessian updating for nonlinear optimization, J. appl. math. comput., 29, 67-80, (2009) · Zbl 1172.49019
[8] Nocedal, J.; Wright, S., Numerical optimization, (1999), Springer Verlag New York, NY · Zbl 0930.65067
[9] Long, J.; Zeng, S.Y., A projection filter method for solving nonlinear complementarity problems, Appl. math. comput., 216, 300-307, (2010) · Zbl 1192.65080
[10] Nie, P.Y., A null space method for solving system of equations, Appl. math. comput., 149, 215-226, (2004) · Zbl 1044.65041
[11] Nie, P.Y., An SQP approach with line search for a system of nonlinear equations, Math. comput. model., 43, 368-373, (2006) · Zbl 1171.90551
[12] Nie, P.Y.; Lai, M.Y.; Zhu, S.J.; Zhang, P.A., A line search filter approach for the system of nonlinear equations, Comput. math. appl., 55, 2134-2141, (2008) · Zbl 1144.90491
[13] Nie, P.Y., A derivative-free method for the system of nonlinear equations, Nonlinear anal. real world appl., 7, 378-384, (2006) · Zbl 1120.90060
[14] Nie, P.Y., CDT like approach for the system of equations, Appl. math. comput., 172, 892-902, (2006) · Zbl 1088.65049
[15] Nie, P.Y., A filter method for solving nonlinear complementarity problems, Appl. math. comput., 167, 677-694, (2005) · Zbl 1082.65062
[16] Nie, P.Y., Sequential penalty quadratic programming filter methods for nonlinear programming, Nonlinear anal. real world appl., 8, 118-129, (2007) · Zbl 1168.90018
[17] Powell, M., A hybrid method for nonlinear equations, () · Zbl 0277.65028
[18] Shen, C.G.; Xue, W.J.; Pu, D.G., A globally convergent trust region multidimensional filter SQP algorithm for nonlinear programming, Int. J. comput. math., 86, 2201-2217, (2009) · Zbl 1183.65070
[19] Su, K., A globally and superlinearly convergent modified SQP-filter method, J. global optim., 41, 203-217, (2008) · Zbl 1153.90023
[20] Tapia, R., Diagonalized multiplier methods and quasi-Newton methods for constraints optimization, J. optim. theory appl., 22, 135-194, (1977) · Zbl 0336.65034
[21] Ulbrich, M.; Ulbrich, S.; Vicente, L.N., A globally convergent primal-dual interior-point filter method for nonlinear programming, Math. program. ser. A, 100, 379-410, (2004) · Zbl 1070.90110
[22] Wächter, A.; Biegler, L.T., Line search filter methods for nonlinear programming: motivation and global convergence, SIAM J. optim., 16, 1-31, (2005) · Zbl 1114.90128
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.