An efficient line search for nonlinear least squares
Journal of Optimization Theory and Applications
Practical methods of optimization; (2nd ed.)
Practical methods of optimization; (2nd ed.)
Convergence of quasi-Newton matrices generated by the symmetric rank one update
Mathematical Programming: Series A and B
Measures for symmetric rank-one updates
Mathematics of Operations Research
Metric-Based Symmetric Rank-One Updates
Computational Optimization and Applications
Local Convergence of the Symmetric Rank-One Iteration
Computational Optimization and Applications
Iterative solution of nonlinear equations in several variables
Iterative solution of nonlinear equations in several variables
Analysis of a Symmetric Rank-One Trust Region Method
SIAM Journal on Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
A restarting approach for the symmetric rank one update for unconstrained optimization
Computational Optimization and Applications
Improved Hessian approximation with modified secant equations for symmetric rank-one method
Journal of Computational and Applied Mathematics
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Quasi-Newton methods are generally held to be the most efficient minimization methods for small to medium sized problems. From these the symmetric rank one update of Broyden (Math. Comp., vol. 21, pp. 368–381, 1967) has been disregarded for a long time because of its potential failure. The work of Conn, Gould and Toint (Math. Prog., vol. 50, pp. 177–195, 1991), Kelley and Sachs (COAP, vol. 9, pp. 43–64, 1998) and Khalfan, Byrd and Schnabel (SIOPT, vol. 3, pp. 1–24, 1993; SIOPT, vol. 6, pp. 1025–1039, 1996) has renewed the interest in this method. However the question of boundedness of the generated matrix sequence has not been resolved by this work. In the present paper it is shown that a slightly modified version of this update generates bounded updates and converges superlinearly for uniformly convex functions. Numerical results support these theoretical considerations.