## Accumulative approach in multistep diagonal gradient-type method for large-scale unconstrained optimization.(English)Zbl 1254.90226

Summary: We focus on developing diagonal gradient-type methods that employ accumulative approach in multistep diagonal updating to determine a better Hessian approximation in each step. The interpolating curve is used to derive a generalization of the weak secant equation, which will carry the information of the local Hessian. The new parameterization of the interpolating curve in variable space is obtained by utilizing accumulative approach via a norm weighting defined by two positive definite weighting matrices. We also note that the storage needed for all computation of the proposed method is just $$O(n)$$. Numerical results show that the proposed algorithm is efficient and superior by comparison with some other gradient-type methods.

### MSC:

 90C30 Nonlinear programming 65K05 Numerical mathematical programming methods

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 [1] J. Barzilai and J. M. Borwein, “Two-point step size gradient methods,” IMA Journal of Numerical Analysis, vol. 8, no. 1, pp. 141-148, 1988. · Zbl 0638.65055 [2] M. A. Hassan, W. J. Leong, and M. Farid, “A new gradient method via quasi-Cauchy relation which guarantees descent,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 300-305, 2009. · Zbl 1179.65067 [3] W. J. Leong, M. A. Hassan, and M. Farid, “A monotone gradient method via weak secant equation for unconstrained optimization,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 413-423, 2010. · Zbl 1203.90148 [4] W. J. Leong, M. Farid, and M. A. Hassan, “Scaling on diagonal Quasi-Newton update for large scale unconstrained Optimization,” Bulletin of the Malaysian Mathematical Sciences Soceity, vol. 35, no. 2, pp. 247-256, 2012. · Zbl 1246.65092 [5] M. Y. Waziri, W. J. Leong, M. A. Hassan, and M. Monsi, “A new Newtons method with diagonal jacobian approximation for systems of nonlinear equations,” Journal of Mathematics and Statistics, vol. 6, pp. 246-252, 2010. · Zbl 1205.65182 [6] J. E. Dennis, Jr. and H. Wolkowicz, “Sizing and least-change secant methods,” SIAM Journal on Numerical Analysis, vol. 30, no. 5, pp. 1291-1314, 1993. · Zbl 0802.65081 [7] M. Farid, W. J. Leong, and M. A. Hassan, “A new two-step gradient-type method for large-scale unconstrained optimization,” Computers & Mathematics with Applications, vol. 59, no. 10, pp. 3301-3307, 2010. · Zbl 1198.90395 [8] M. Farid and W. J. Leong, “An improved multi-step gradient-type method for large scale optimization,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3312-3318, 2011. · Zbl 1222.90044 [9] J. A. Ford and I. A. Moghrabi, “Alternating multi-step quasi-Newton methods for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 82, no. 1-2, pp. 105-116, 1997. · Zbl 0886.65064 [10] J. A. Ford and S. Tharmlikit, “New implicit updates in multi-step quasi-Newton methods for unconstrained optimisation,” Journal of Computational and Applied Mathematics, vol. 152, no. 1-2, pp. 133-146, 2003. · Zbl 1025.65035 [11] L. Armijo, “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific Journal of Mathematics, vol. 16, pp. 1-3, 1966. · Zbl 0202.46105 [12] R. H. Byrd and J. Nocedal, “A tool for the analysis of quasi-Newton methods with application to unconstrained minimization,” SIAM Journal on Numerical Analysis, vol. 26, no. 3, pp. 727-739, 1989. · Zbl 0676.65061 [13] N. Andrei, “An unconstrained optimization test functions collection,” Advanced Modeling and Optimization, vol. 10, no. 1, pp. 147-161, 2008. · Zbl 1161.90486 [14] J. J. Moré, B. S. Garbow, and K. E. Hillstrom, “Testing unconstrained optimization software,” ACM Transactions on Mathematical Software, vol. 7, no. 1, pp. 17-41, 1981. · Zbl 0454.65049 [15] E. D. Dolan and J. J. Moré, “Benchmarking optimization software with performance profiles,” Mathematical Programming A, vol. 91, no. 2, pp. 201-213, 2002. · Zbl 1049.90004
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