Gradient methods with adaptive step-sizes. (English) Zbl 1121.90099

Summary: Motivated by the superlinear behavior of the Barzilai-Borwein (BB) method for two-dimensional quadratics, we propose two gradient methods which adaptively choose a small step-size or a large step-size at each iteration. The small step-size is primarily used to induce a favorable descent direction for the next iteration, while the large step-size is primarily used to produce a sufficient reduction. Although the new algorithms are still linearly convergent in the quadratic case, numerical experiments on some typical test problems indicate that they compare favorably with the BB method and some other efficient gradient methods.


90C20 Quadratic programming
90C52 Methods of reduced gradient type


Full Text: DOI


[1] 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. · Zbl 0100.14002
[2] R.Andreani, E.G.Birgin, J.M.Martínez, and J.Yuan, ”Spectral projected gradient and variable metric methods for optimization with linear inequalities,” IMA J.Numer. Anal., vol. 25, pp. 221–252, 2005. · Zbl 1072.90051
[3] J.Barzilai and J.M.Borwein, ”Two-point step size gradient methods,” IMA J. Numer. Anal., vol. 8, pp. 141–148, 1988. · Zbl 0638.65055
[4] L.Bello and M.Raydan, ”Preconditioned spectral projected gradient methods on convex sets,” Journal of Computational Mathematics, vol. 23, pp. 225–232, 2005. · Zbl 1110.65048
[5] E.G.Birgin, J.M.Martínez, and M.Raydan, ”Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim., vol. 10, pp. 1196–1211, 2000. · Zbl 1047.90077
[6] E.G.Birgin, J.M.Martínez, and M.Raydan, ”Algorithm 813: SPG–software for convex-constrained optimization,” ACM Trans. Math. Software, vol. 27, pp. 340–349, 2001. · Zbl 1070.65547
[7] E.G.Birgin, J.M.Martínez, and M.Raydan, ”Inexact spectral projected gradient methods on convex sets,” IMA J. Numer. Anal., vol. 23, pp. 539–559, 2003. · Zbl 1047.65042
[8] A.Cauchy, ”Méthode générale pour la résolution des syst è ms d’équations simultanées,” Comp. Rend. Sci. Paris, vol. 25, pp. 536–538, 1847.
[9] Y.H.Dai, ”Alternate step gradient method,” Optimization, vol. 52, pp. 395–415, 2003. · Zbl 1056.65055
[10] Y.H.Dai and L.Z.Liao, ”R-linear convergence of the Barzilai and Borwein gradient method,” IMA J. Numer. Anal., vol. 22, pp. 1–10, 2002. · Zbl 1002.65069
[11] Y.H.Dai and Y.Yuan, ”Alternate minimization gradient method,” IMA J. Numer. Anal., vol. 23, pp. 377–393, 2003. · Zbl 1055.65073
[12] R.Fletcher, ”Low storage methods for unconstrained optimization,” Lectures in Applied Mathematics (AMS), vol. 26, pp. 165–179, 1990. · Zbl 0699.65052
[13] R.Fletcher, ”On the Barzilai-Borwein method,” Numerical Analysis Report NA/207, Department of Mathematics, University of Dundee, 2001. · Zbl 1118.90318
[14] 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. · Zbl 0940.65032
[15] L.Grippo, F.Lampariello, and S.Lucidi, ”A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal., vol. 23, pp. 707–716, 1986. · Zbl 0616.65067
[16] M.R.Hestenes and E.L.Stiefel, ”Methods of conjugate gradients for solving linear systems,” J. Research National Bureau of Standards, vol. B49, pp. 409–436, 1952. · Zbl 0048.09901
[17] W.LaCruz, J.M.Martínez, and M.Raydan, ”Spectral residual method without gradient information for solving large-scale nonlinear systems of equations,” Mathematics of Computation, to appear.
[18] W.LaCruz and M.Raydan, ”Nonmonotone spectral methods for large-scale nonlinear systems,” Optimization Methods and Software, vol. 18, pp. 583–599, 2003. · Zbl 1069.65056
[19] J.-L.Lamotte, B.Molina, and M.Raydan, ”Smooth and adaptive gradient method with retards,” Mathematical and Computer Modelling, vol. 36, pp. 1161–1168, 2002. · Zbl 1029.65029
[20] F.Luengo and M.Raydan, ”Gradient method with dynamical retards for large-scale optimization problems,” Electronic Transactions on Numerical Analysis, vol. 16, pp. 186–193, 2003. · Zbl 1134.90496
[21] M.Raydan, ”On the Barzilai and Borwein choice of steplength for the gradient method,” IMA J. Numer. Anal., vol. 13, pp. 321–326, 1993. · Zbl 0778.65045
[22] M.Raydan, ”The Barzilai and Borwein method for the large scale unconstrained minimization problem,” SIAM J. Optim., vol. 7, pp. 26–33, 1997. · Zbl 0898.90119
[23] M.Raydan and B.F.Svaiter, ”Relaxed steepest descent and Cauchy-Barzilai-Borwein method,” Computational Optimization and Applications, vol. 21, pp. 155–167, 2002. · Zbl 0988.90049
[24] T.Serafini, G.Zanghirati, and L.Zanni, ”Gradient projection methods for quadratic programs and applications in training support vector machines,” Tech. Rep. 48, University of Modena and Reggio Emilia, Italy, 2003. · Zbl 1054.65062
[25] H.Zhang and W.W. Hager, ”A nonmonotone line search technique and its application to unconstrained optimization,” SIAM J. Optim., vol. 14, pp. 1043–1056, 2004. · Zbl 1073.90024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.