Abstract
We consider conditions under which the SR1 iteration is locally convergent. We apply the result to a pointwise structured SR1 method that has been used in optimal control.
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References
C.G. Broyden, “Quasi-Newton methods and their application to function minimization,” Math. Comp., vol. 21, pp. 368-381, 1967.
C.G. Broyden, J.E. Dennis, and J.J. Moré, “On the local and superlinear convergence of quasi-Newton methods,” J. Inst. Maths. Applics., vol. 12, pp. 223-246, 1973.
R.H. Byrd, H.F. Khalfan, and R.B. Schnabel, “Analysis of a symmetric rank-one trust region method,” Tech. Report CU-CS-657-93, University of Colorado at Boulder, Department of Computer Science, July 1993. SIAM J. Opt. (to appear).
A.R. Conn, N.I.M. Gould, and P.L. Toint, “Testing a class of methods for solving minimization problems with simple bounds on the variables,” Math. Comp., vol. 50, pp. 399-430, 1988.
_____, “Convergence of quasi-Newton matrices generated by the symmetric rank one update,” Math. Programming A, vol. 50, pp. 177-195, 1991.
J.E. Dennis and R.B. Schnabel, Numerical Methods for Nonlinear Equations and Unconstrained Optimization, Prentice-Hall: Englewood Cliffs, N.J., 1983.
J.E. Dennis and H.F. Walker, “Convergence theorems for least change secant update methods,” SIAM J. Numer. Anal., vol. 18, pp. 949-987, 1981.
P. Deuflhard, R.W. Freund, and A. Walter, “Fast secant methods for the iterative solution of large non-symmetric linear systems,” Impact of Computing in Science and Engineering, vol. 2, pp. 244-276, 1990.
A.V. Fiacco and G.P. McCormick, Nonlinear Programming, John Wiley and Sons: New York, 1968.
P. Gilmore, “An algorithm for optimizing functions with multiple minima,” Ph.D. Thesis, North Carolina State University, 1993.
_____, “IFFCO: Implicit filtering for constrained optimization,” Tech. Report CRSC-TR93-7, Center for Research in Scientific Computation, North Carolina State University, May 1993. Available by anonymous ftp from math.ncsu.edu in pub/kelley/iffco/ug.ps.
H.B. Keller, “Numerical solution of two point boundary value problems,” CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1982.
C.T. Kelley, “Iterative methods for linear and nonlinear equations,” No. 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia, 1995.
C.T. Kelley, E.W. Sachs, and B. Watson, “A pointwise quasi-Newton method for unconstrained optimal control problems,” II, J. Optim. Theory Appl., vol. 71, pp. 535-547, 1991.
H. Khalfan, R.H. Byrd, and R.B. Schnabel, “A theoretical and experimental study of the symmetric rank one update,” SIAM J. Optimization, vol. 3, pp. 1-24, 1993.
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press: New York, 1970.
M.R. Osborne and L.P. Sun, “A new approach to the symmetric rank-one updating algorithm,” Tech. Report, Department of Statistics, IAS, Australian National University, 1988.
D. Stoneking, G. Bilbro, R. Trew, P. Gilmore, and C.T. Kelley, “Yield optimization using a GaAs process simulator coupled to a physical device model,” IEEE Transactions on Microwave Theory and Techniques, vol. 40, pp. 1353-1363, 1992.
P. L. Toint, “On large scale nonlinear least squares calculations,” SIAM J. Scientific and Statistical Computing, vol. 8, pp. 416-435, 1987.
B. Watson, “Quasi-Newton methoden für Minimierungsprobleme mit strukturierter Hesse-matrix,” Diploma Thesis, Universität Trier, 1990.
S.J. Wright, “Compact storage of Broyden-class quasi-Newton matrices,” Argonne National Laboratory, 1994 (preprint).
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Kelley, C., Sachs, E. Local Convergence of the Symmetric Rank-One Iteration. Computational Optimization and Applications 9, 43–63 (1998). https://doi.org/10.1023/A:1018330119731
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DOI: https://doi.org/10.1023/A:1018330119731