Abstract
The Barzilai and Borwein gradient method does not ensure descent in the objective function at each iteration, but performs better than the classical steepest descent method in practical computations. Combined with the technique of nonmonotone line search etc., such a method has found successful applications in unconstrained optimization, convex constrained optimization and stochastic optimization. In this paper, we give an analysis of the Barzilai and Borwein gradient method for two unsymmetric linear equations with only two variables. Under mild conditions, we prove that the convergence rate of the Barzilai and Borwein gradient method is Q-superlinear if the coefficient matrix A has the same eigenvalue; if the eigenvalues of A are different, then the convergence rate is R-superlinear.
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References
H. Akaike (1959), 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.
J. Barzilai and J. M. Borwein (1988), Two-point step size gradient methods, IMA J. Numer. Anal., Vol. 8, pp. 141–148.
E. G. Birgin, I. Chambouleyron, and J. M. MartÃnez (1999), Estimation of the optical constants and the thickness of thin films using unconstrained optimization, J. Comput. Phys., Vol. 151, pp. 862–880.
E. G. Birgin and Y. G. Evtushenko (1998), Automatic differentiation and spectral projected gradient methods for optimal control problems, Optim. Methods Softw., Vol. 10, pp. 125–146.
E. G. Birgin, J. M. MartÃnez, and M. Raydan (2000), Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optim., Vol. 10, pp. 1196–1211.
A. Cauchy (1847), Méthode générale pour la résolution des systèmes d'équations simultanées, Comp. Rend. Acad. Sci. Paris, Vol. 25, pp. 46–89.
Y. H. Dai and L.-Z. Liao (2002), R-linear convergence of the Barzilai and Borwein gradient method, IMA J. Numer. Anal., Vol. 22, No. 1, pp. 1–10.
R. Fletcher (1990), Low storage methods for unconstrained optimization, Lectures in Applied Mathematics (AMS), Vol. 26, pp. 165–179.
A. Friedlander, J. M. MartÃnez, B. Molina, and M. Raydan (1999), Gradient method with retards and generalizations, SIAM J. Numer. Anal., Vol. 36, 275–289.
W. Glunt, T. L. Hayden, and M. Raydan (1993), Molecular conformations from distance matrices, J. Comput. Chem., Vol. 14, pp. 114–120.
L. Grippo, F. Lampariello, and S. Lucidi (1986), A nonmonotone line search technique for Newton's method, SIAM J. Numer. Anal., Vol. 23, pp. 707–716.
W. B. Liu and Y. H. Dai (2001), Minimization Algorithms based on Supervisor and Searcher Cooperation, Journal of Optimization Theory and Applications, Vol. 111, No. 2, pp. 359–379.
M. Raydan (1993), On the Barzilai and Borwein choice of steplength for the gradient method, IMA J. Numer. Anal., Vol. 13, pp. 321–326.
M. Raydan (1997), The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., Vol. 7, pp. 26–33.
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Dai, YH., Liao, LZ., Li, D. (2005). An Analysis of the Barzilai and Borwein Gradient Method for Unsymmetric Linear Equations. In: Qi, L., Teo, K., Yang, X. (eds) Optimization and Control with Applications. Applied Optimization, vol 96. Springer, Boston, MA. https://doi.org/10.1007/0-387-24255-4_8
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DOI: https://doi.org/10.1007/0-387-24255-4_8
Publisher Name: Springer, Boston, MA
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