Convergence of quasiNewton matrices generated by the symmetric rank one update
 A. R. Conn,
 N. I. M. Gould,
 Ph. L. Toint
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QuasiNewton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives. This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used. The rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates. The theory is confirmed by some numerical experiments that also show the convergence of the Hessian approximations to be substantially slower for other known quasiNewton formulae.
 A.R. Conn, N.I.M. Gould and Ph.L. Toint, “Global convergence of a class of trust region algorithms for optimization with simple bounds,”SIAM Journal on Numerical Analysis 25 (1988) 433–460 (with a correction given inSIAM Journal on Numerical Analysis 26 (1989) 764–767).
 A.R. Conn, N.I.M. Gould and Ph.L. Toint, “Testing a class of methods for solving minimization problems with simple bounds on the variables,”Mathematics of Computation 50 (1988) 399–430.
 J.E. Dennis and J.J. Moré, “QuasiNewton methods, motivation and theory,”SIAM Review 19 (1977) 46–89.
 J.E. Dennis and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (PrenticeHall, Englewood Cliffs, NJ, 1983).
 A.V. Fiacco and G.P. McCormick,Nonlinear Programming (Wiley, New York, 1968).
 R. Fletcher,Practical Methods of Optimization: Unconstrained Optimization (Wiley, Chichester, 1980).
 R.P. Ge and M.J.D. Powell, “The convergence of variable metric matrices in unconstrained optimization,”Mathematical Programming 27 (1983) 123–143.
 P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, New York, 1981).
 A. Griewank and Ph.L. Toint, “Partitioned variable metric updates for large structured optimization problems,”Numerische Mathematik 39 (1982) 119–137.
 J.J. Moré, “Recent developments in algorithms and software for trust region methods,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art (Springer, Berlin, 1983).
 J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
 M.J.D. Powell, “A new algorithm for unconstrained optimization,” in: J.B. Rosen, O.L. Mangasarian and K. Ritter, eds.,Nonlinear Programming (Academic Press, New York, 1970).
 G. Schuller, “On the order of convergence of certain quasiNewton methods,”Numerische Mathematik 23 (1974) 181–192.
 D.C. Sorensen, “An example concerning quasiNewton estimates of a sparse Hessian,”SIGNUM Newsletter 16 (1981) 8–10.
 Ph.L. Toint, “On the superlinear convergence of an algorithm for solving a sparse minimization problem,”SIAM Journal on Numerical Analysis 16 (1979) 1036–1045.
 Title
 Convergence of quasiNewton matrices generated by the symmetric rank one update
 Journal

Mathematical Programming
Volume 50, Issue 13 , pp 177195
 Cover Date
 19910301
 DOI
 10.1007/BF01594934
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 QuasiNewton updates
 convergence theory
 Industry Sectors
 Authors

 A. R. Conn ^{(1)}
 N. I. M. Gould ^{(2)}
 Ph. L. Toint ^{(3)}
 Author Affiliations

 1. IBM T. J. Watson Research Center, 10598, Yorktown Heights, NY, USA
 2. Rutherford Appleton Laboratory, Chilton, UK
 3. Department of Mathematics, Facultés Universitaires ND de la Paix, B5000, Namur, Belgium