Convergence of quasi-Newton method with new inexact line search. (English) Zbl 1093.65063

The author proposes a new inexact line search rule for a quasi-Newton method and establishes some global convergence results of this method. These results are useful in designing new quasi-Newton methods. Also, there is analysed the convergence rate of the quasi-Newton method with the new line search rule, is analyzed.


65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C53 Methods of quasi-Newton type
Full Text: DOI


[1] Armijo, L., Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. math., 16, 1-3, (1966) · Zbl 0202.46105
[2] Goldstein, A.A., On steepest descent, SIAM J. control, 3, 147-151, (1965) · Zbl 0221.65094
[3] Nocedal, J.; Wright, J.S., Numerical optimization, (1999), Springer-Verlag New York · Zbl 0930.65067
[4] Polak, E., Optimization: algorithms and consistent approximations, (1997), Springer-Verlag New York · Zbl 0899.90148
[5] Shi, Z.-J., Convergence of line search methods for unconstrained optimization, Appl. math. comput., 154, 393-405, (2004) · Zbl 1072.65087
[6] Todd, M.J., On convergence properties of algorithms for unconstrained minimization, IMA J. numer. anal., 9, 435-441, (1989) · Zbl 0686.65032
[7] Wei, Z.; Qi, L.; Jiang, H., Some convergence properties of descent methods, J. optim. theory appl., 95, 177-188, (1997) · Zbl 0890.90167
[8] Wolfe, P., Convergence condition for ascent methods II: some corrections, SIAM rev., 13, 185-188, (1971) · Zbl 0216.26901
[9] Wolfe, P., Convergence conditions for ascent methods, SIAM rev., 11, 226-235, (1969) · Zbl 0177.20603
[10] Vrahatis, M.N.; Androulakis, G.S.; Lambrinos, J.N.; Magoulas, G.D., A class of gradient unconstrained minimization algorithms with adaptive step size, J. comput. appl. math., 114, 367-386, (2000) · Zbl 0958.65072
[11] Vrahatis, M.N.; Androulakis, G.S.; Manoussakis, G.E., A new unconstrained optimization method for imprecise function and gradient values, J. math. anal. appl., 197, 586-607, (1996) · Zbl 0887.90166
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