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An affine scaling trust-region approach to bound-constrained nonlinear systems. (English) Zbl 1018.65067
The paper presents an iterative method for numerically finding zero points of nonlinear maps which lie within a given cell. The method combines ideas from trust-region Newton methods and an affine interior scaling approach for constrained optimization. Global and local convergence properties are derived. Extensive numerical tests on a large class of test problems from the literature are reported. The present method appears to compare well with an active set-type Newton method and an inexact Gauss-Newton-type method.

65H10 Numerical computation of solutions to systems of equations
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[1] Conn, A.R.; Gould, N.I.M.; Toint, Ph.L., LANCELO: A Fortran package for large-scale nonlinear optimization, Springer ser. comput. math., 17, (1992), Springer New York · Zbl 0809.65066
[2] Coleman, T.F.; Li, Y., On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds, Math. programming, 67, 189-224, (1994) · Zbl 0842.90106
[3] Coleman, T.F.; Li, Y., An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. optim., 6, 418-445, (1996) · Zbl 0855.65063
[4] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
[5] Dennis, J.E.; Vicente, L.N., Trust-region interior-point algorithms for minimization problems with simple bounds, (), 97-107 · Zbl 0907.65056
[6] Ferris, M.C.; Pang, J.S., Engineering and economic applications of complementarity problems, SIAM rev., 39, 669-713, (1997) · Zbl 0891.90158
[7] Floudas, C.A., Handbook of test problems in local and global optimization, Nonconvex optimization and its applications, 33, (1999), Kluwer Academic Dordrecht
[8] Hock, W.; Schittkowski, K., Test examples for nonlinear programming codes, Lecture notes in econom. and math. systems, 187, (1981), Springer Berlin
[9] Kanzow, C., An active set-type Newton method for constrained nonlinear systems, (), 179-200 · Zbl 0983.90060
[10] Kanzow, C., Strictly feasible equation-based methods for mixed complementarity problems, Numer. math., 89, 135-160, (2001) · Zbl 0992.65070
[11] Kelley, C.T., Iterative methods for solving linear and nonlinear equations, Frontiers in applied mathematics, 16, (1995), SIAM Philadelphia, PA · Zbl 0832.65046
[12] Kelley, C.T., Iterative methods for optimization, Frontiers in applied mathematics, 18, (1999), SIAM Philadelphia, PA · Zbl 0934.90082
[13] Kozakevich, D.N.; Martinez, J.M.; Santos, S.A., Solving nonlinear systems of equations with simple bounds, Comput. appl. math., 16, 215-235, (1997) · Zbl 0896.65041
[14] S.P. Dirkse, M.C. Ferris, MCPLIB: A collection of nonlinear mixed complementary problems, Technical Report, Computer Sciences Department, University of Wisconsin, Madison, WI, 1994
[15] Lin, C.J.; Moré, J.J., Newton’s method for large bound-constrained optimization problems, SIAM J. optim., 9, 1100-1127, (1999) · Zbl 0957.65064
[16] Meintjes, K.; Morgan, A.P., Chemical equilibrium systems as numerical tests problems, ACM trans. math. software, 16, 143-151, (1990) · Zbl 0900.65153
[17] Meintjes, K.; Morgan, A.P., A methodology for solving chemical equilibrium systems, Appl. math. comput., 22, 333-361, (1987) · Zbl 0616.65057
[18] Moré, J.J., A collection of nonlinear model problems, (), 723-762
[19] Moré, J.J.; Sorensen, D.C., Computing a trust region step, SIAM J. sci. statist. comput., 4, 553-572, (1983) · Zbl 0551.65042
[20] Nocedal, J.; Wright, S.J., Numerical optimization, Springer series in operations research, (1999), Springer New York · Zbl 0930.65067
[21] Sartenaer, A., Automatic detection of an initial trust region in nonlinear programming, SIAM J. sci. comput., 18, 1788-1803, (1997) · Zbl 0891.90151
[22] Ulbrich, M., Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems, SIAM J. optim., 11, 889-917, (2000) · Zbl 1010.90085
[23] Wang, T.; Monteiro, R.D.C.; Pang, K.J.S., A potential reduction Newton method for constrained equations, Math. programming, 74, 159-195, (1996)
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