Abstract
For unconstrained optimization, the two-point stepsize gradient method is preferable over the classical steepest descent method both in theory and in real computations. In this paper we interpret the choice for the stepsize in the two-point stepsize gradient method from the angle of interpolation and propose two modified two-point stepsize gradient methods. The modified methods are globally convergent under some mild assumptions on the objective function. Numerical results are reported, which suggest that improvements have been achieved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Akaike, “On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method,” Ann. Inst. Statist. Math. Tokyo, vol. 11, pp. 1–17, 1959.
J. Barzilai and J.M. Borwein, “Two point step size gradient methods,” IMA J. Numer. Anal., vol. 8, pp. 141–148, 1988.
E.G. Birgin, I. Chambouleyron, and J.M. Martínez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys., vol. 151, pp. 862–880, 1999.
E.G. Birgin and Y.G. Evtushenko, “Automatic differentiation and spectral projected gradient methods for optimal control problems,” Optim. Methods Softw., vol. 10, pp. 125–146, 1998.
E.G. Birgin, J.M. Martínez, and M. Raydan, “Nonmonotone spectral projected gradient methods for convex sets,” SIAM Journal on Optimization, vol. 10, no. 4, pp. 1196–1211, 2000.
A. Cauchy, “Méthode générale pour la résolution des systèms d'equations simultanées,” Comp. Rend. Sci. Paris, vol. 25, pp. 46–89, 1847.
Y.H. Dai and L.Z. Liao, “R-Linear Convergence of the Barzilai and Borwein Gradient Method,” Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Research report AMSS-1999-081, 1999. Also in IMA J. Numer. Anal., accepted.
R. Fletcher, “Low storage methods for unconstrained optimization,” Lectures in Applied Mathematics (AMS), vol. 26, pp. 165–179, 1999.
G.E. Forsythe, “On the asymptotic directions of the s-dimensional optimum gradient method,” Numerische Mathematik, vol. 11, pp. 57–76, 1968.
A. Friedlander, J.M. Martínez, B. Molina, and M. Raydan, “Gradient method with retards and generalizations,” SIAM J. Numer. Anal., vol. 36, pp. 275–289, 1999.
W. Glunt, T.L. Hayden, and M. Raydan, “Molecular conformations from distance matrices,” J. Comput. Chem., vol. 14, pp. 114–120, 1993.
L. Grippo, F. Lampariello, and S. Lucidi, “Anonmonotone line search technique for Newton's method,” SIAM J. Numer. Anal., vol. 23, pp. 707–716, 1986.
W.B. Liu and Y.H. Dai, “Minimization Algorithms based on Supervisor and Searcher Co-operation. I:-Faster and robust gradient algorithms for minimization problems with stronger noises,” Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Research report AMSS-1999-085, 1999. Also in JOTA, accepted.
J.J. Morè, B.S. Garbow, and K.E. Hillstrom, “Testing unconstrained optimization software,”ACMTransactions on Mathematical Software, vol. 7, pp. 17–41, 1981.
M. Raydan, “On the Barzilai and Borwein choice of steplength for the gradient method,” IMA J. Numer. Anal., vol. 13, pp. 321–326, 1993.
M. Raydan, “The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,” SIAM J. Optim., vol. 7, no. 1, pp. 26–33, 1997.
Y. Yuan “Amodified BFGS algorithm for unconstrained optimization,” IMA J. Numer. Anal., vol. 11, pp. 325–332, 1991.
Y. Yuan, “Numerical methods for nonlinear programming,” Shanghai Scientific and Technical Publishers, 1993 (in Chinese).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dai, Y., Yuan, J. & Yuan, YX. Modified Two-Point Stepsize Gradient Methods for Unconstrained Optimization. Computational Optimization and Applications 22, 103–109 (2002). https://doi.org/10.1023/A:1014838419611
Issue date:
DOI: https://doi.org/10.1023/A:1014838419611
