Modification of the Armijo line search to satisfy the convergence properties of HS method

[1] Armijo, L., Minimization of functions having
Lipschits continuous partial derivatives, Pacific
J. Math., 16, (1966) 1–3.
[2] Al-Baali, M., Descent property and global
convergence of the Fletcher–Reeves method
with inexact line search, IMA J. Numer.
Anal., 5, 121–124 (1985).
[3] Birgin, E.G., Martinez, J.M., A spectral conjugate
gradient method for unconstrained optimization,
Appl. Math. Optim., 43, 117–128
(2001).
[4] Chen, X.D., Sun, J., Global convergence of
a two-parameter family of conjugate gradient
methods without line search, Journal
of Computational and Applied Mathematics,
146, 37–45 (2002).
[5] Dai, Y.H., Yuan, Y., Convergence properties
of the conjugate descent method, Advances
in Mathematics, 25, 552–562 (1996).
[6] Dai, Y.H., Yuan, Y., A nonlinear conjugate
gradient method with a strong global convergence
property, SIAM Journal of Optimization,
10, 177–182 (1999).
[7] Dai, Y.H., Han, J.Y., Liu, G.H., Sun, D.F.,
Yin, H.X., Yuan, Y.X., Convergence properties
of nonlinear conjugate gradient methods,
SIAM Journal of Optimization, 10, 345–358
(1999).
[8] Fletcher, R., Practical Method of Optimization,
second ed., vol. I: Unconstrained Optimization,
Wiley, New York, (1997).
[9] Fletcher, R., Reeves, C., Function minimization
by conjugate gradients, Comput. J., 7,
149–154 (1964).
[10] Fasano, G., Planar conjugate gradient algorithm
for large scale unconstrained optimization.
Part 1: Theory, Journal of Optimization
Theory and Applications, 125, 523–541
(2005).
[11] Fasano, G., Planar conjugate gradient algorithm
for large-scale unconstrained optimization.
Part 1: Applications, Journal of Optimization
Theory and Applications, 125, 543–
558 (2005).
[12] Grippo, L., Lucidi, S., A globally convergent
version of the Polak–Ribi`ere conjugate gradient
method, Mathematical Programming, 78,
375–391 (1997).
[13] Grippo, L., Lucidi, S., Convergence conditions,
line search algorithms and trust region
implementations for the Polak Ribi`ere conjugate
gradient method, Optimization Methods
and Software, 20(1), 71–98 (2005).
[14] Goldstein, A.A., On steepest descent, SIAM
J. Control, 3, 147–151 (1965).
[15] Hestenes, M.R., and Stiefel, E.L., Methods
of conjugate gradients for solving linear systems,
J. Res. Nat. Bur. Standars Sect., 5(49),
409-436 (1952).
[16] Hu, Y.F., Storey, C., Global convergence result
for conjugate gradient methods, Journal
of Optimization Theory and Applications, 71,
399–405 (1991).
[17] Liu, Y., Storey, C., Efficient generalized conjugate
gradient algorithms. Part 1: Theory,
Journal of Optimization Theory and Applications,
69, 129–137 (1991).
[18] Mor´e, J.J., Garbow, B.S., Hillstrom, K.E.,
Testing unconstrained optimization software,
ACM Trans. Math. Softw., 7, 17–41 (1981).
[19] Polak, E., Ribi`ere, G., Note Sur la convergence
de directions conjug`ees, Rev. Francaise
Informat Recherche Operationelle, 3e
Ann`ee16, 35–43 (1969).
[20] Polyak, B.T., The conjugate gradient
method in extreme problems, USSR Comp.
Math. Math. Phys., 9, 94–112 (1969).
[21] Powell, M.J.D., Nonconvex minimization
calculations and the conjugate gradient
method, Lecture Notes in Mathematics,
1066, Springer-Verlag, Berlin, 122–141
(1984).
[22] Sun, J., Zhang, J.P., Convergence of conjugate
gradient methods without line search,
Annals of Operations Research, 103, 161–173
(2001).
[23] Sun, J., Yang, X.Q., Chen, X.D., Quadratic
cost flow and the conjugate gradient method,
European Journal of Operational Research,
164, 104–114 (2005).
[24] Shi, Z.J., Shen, J., Convergence of descent
method without line search, Appl. Math.
Comput., 167, 94–107 (2005).
[25] Shi, Z.J., Restricted PR conjugate gradient
method and its global convergence, Advances
in Mathematics, 31(1), 47–55 (2002)(in Chinese).
[26] Shi, Z.J., Shen, J., Convergence of Polak
Rebi`ere Polyak conjugate gradient method,
Nonlinear Analysis, 66, 1428–1441 (2007).
[27] Shi, Z.J., Shen, J., Convergence of LiuStorey
conjugate gradient method, European
Journal of Operational Research,182(2), 552–
560 (2007).
[28] Wolfe, P. Convergence conditions for ascent
methods, SIAM Rev., 11, 226–235 (1969).
[29] Yuan, Y. Numerical Methods for Nonlinear
Programming, Shanghai Scientific & Technical
Publishers, (1993).
[30] Yuan, Y. Analysis on the conjugate gradient
method, Optimization Methods and Software,
2, 19–29 (1993).
[31] Zoutendijk, G. Nonlinear programming computational
methods, in: J. Abadie (Ed.), Integer
and Nonlinear Programming, NorthHolland,
Amsterdam, 37–86 (1970).