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A discrete Newton algorithm for minimizing a function of many variables.(English)Zbl 0477.90055


MSC:

90C30 Nonlinear programming
49M15 Newton-type methods
65K05 Numerical mathematical programming methods
68Q25 Analysis of algorithms and problem complexity

Cite

References:

[1]W. Murray,Numerical methods for unconstrained optimization (Academic Press, New York, 1972). ·Zbl 0264.49026
[2]D.P. O’Leary, ”A discrete Newton algorithm for minimizing a function of many variables”, Computer Science Department Report TR-910, University of Maryland (June 1980).
[3]R.P. Brent, ”Some efficient algorithms for solving systems of nonlinear equations”,SIAM Journal on Numerical Analysis 10 (1973) 327–344. ·Zbl 0258.65051 ·doi:10.1137/0710031
[4]A.R. Curtis, M.J.D. Powell and J.K. Reid, ”On the estimation of sparse Jacobian matrices”,Journal of the Institute of Mathematics and its Applications 13 (1974) 117–119. ·Zbl 0273.65036
[5]M.J.D. Powell and Ph.L. Toint, ”On the estimation of sparse Hessian matrices”,SIAM Journal on Numerical Analysis 16 (1979) 1060–1074. ·Zbl 0426.65025 ·doi:10.1137/0716078
[6]M.C. Bartholomew-Biggs, ”A matrix modification method for calculating approximate solutions to systems of linear equations”,Journal of the Institute of Mathematics and its Applications 23 (1979) 131–137. ·Zbl 0397.65023 ·doi:10.1093/imamat/23.2.131
[7]P.E. Gill and W. Murray, ”Newton-type methods for unconstrained and linearly constrained optimization”,Mathematical Programming 7 (1974) 311–350. ·Zbl 0297.90082 ·doi:10.1007/BF01585529
[8]S. Kaniel and A. Dax, ”A modified Newton’s method for unconstrained minimization”,SIAM Journal on Numerical Analysis 16 (1979) 324–331. ·Zbl 0403.65027 ·doi:10.1137/0716024
[9]P.E. Gill and W. Murray, ”Newton-type methods for linearly constrained optimization”, in: P.E. Gill and W. Murray, eds.,Numerical methods for constrained optimization (Academic Press, New York, 1974) pp. 29–66. ·Zbl 0297.90082
[10]A.A. Goldstein, ”On steepest descent”,SIAM Journal on Control 3 (1965) 147–151. ·Zbl 0221.65094
[11]M.R. Hestenes and E. Stiefel, ”Methods of conjugate gradients for solving linear systems”,Journal of Research of the National Bureau of Standards 49 (1952) 409–436. ·Zbl 0048.09901
[12]C. Lanczos, ”An iteration method for the solution of the eigenvalue problem of linear differential and integral operators”,Journal of Research of the National Bureau of Standards 45 (1950) 255–282.
[13]C. Lanczos, ”Solution of systems of linear equations by minimized iterations”,Journal of Research of the National Bureau of Standards 49 (1952) 33–53.
[14]Rati Chandra, ”Conjugate gradient methods for partial differential equations”, Ph.D. Thesis, Report 129, Department of Computer Science, Yale University (1978).
[15]C.C. Paige and M.A. Saunders, ”Solutions of sparse indefinite systems of linear equations”,SIAM Journal on Numerical Analysis 12 (1975) 617–629. ·Zbl 0319.65025 ·doi:10.1137/0712047
[16]J.R. Bunch and B.N. Parlett, ”Direct methods for solving symmetric indefinite systems of linear equations”,SIAM Journal on Numerical Analysis 8 (1971) 639–655. ·doi:10.1137/0708060
[17]J.W. Daniel, ”The conjugate gradient method for linear and nonlinear operator equations”,SIAM Journal on Numerical Analysis 4 (1967) 10–26. ·Zbl 0154.40302 ·doi:10.1137/0704002
[18]S. Kaniel, ”Estimates for some computational techniques in linear algebra”,Mathematics of Computation 20 (1966) 369–378. ·Zbl 0156.16202 ·doi:10.1090/S0025-5718-1966-0234618-4
[19]O. Axelsson, ”Solution of linear systems of equations: Iterative methods”, in: V.A. Barker, ed.,Sparse matrix techniques (Springer-Verlag, New York, 1977). ·Zbl 0354.65021
[20]P. Concus, G.H. Golub and D.P. O’Leary, ”A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations”, in: J.R. Bunch and D.J. Rose, eds.,Sparse matrix computations (Academic Press, New York, 1976) pp. 309–332.
[21]M.R. Hestenes,Conjugate direction methods in optimization (Springer-Verlag, New York, 1980). ·Zbl 0439.49001
[22]R.F. Dennemeyer and E.H. Mookini, ”CGS algorithms for unconstrained minimization of functions”,Journal of Optimization Theory and Applications 16 (1975) 67–85. ·Zbl 0281.65043 ·doi:10.1007/BF00935624
[23]J.M. Ortega and W.C. Rheinboldt,Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970). ·Zbl 0241.65046
[24]J.C.P. Bus, ”Convergence of Newton-like methods for solving systems of nonlinear equations”,Numerische Mathematik 27 (1977) 271–281. ·Zbl 0332.65032 ·doi:10.1007/BF01396177
[25]P.E. Gill and W. Murray, ”Conjugate gradient methods for large-scale nonlinear optimization”, Systems Optimization Lab Report SOL-79-15, Department of Operations Research, Stanford University (1979).
[26]M.R. Osborne, ”An efficient weak line search with guaranteed termination”, Report 1870, Mathematics Research Center, University of Wisconsin (1978).
[27]N.K. Garg and R.A. Tapia, ”QDN: A variable storage algorithm for unconstrained optimization”, Department of Mathematical Sciences Report, Rice University (1980).
[28]R.S. Dembo and T. Steihaug, ”Truncated-Newton algorithms for large-scale unconstrained optimization”, School of Organization and Management, Yale University Preliminary Draft (September 1980). ·Zbl 0523.90078
[29]P.E. Gill, W. Murray and S.G. Nash, ”Newton-type minimization using the linear conjugate gradient method”, Draft, Department of Operations Research, Stanford University (October 1980).
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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