Abstract
In this paper, a new trust region subproblem is proposed. The trust radius in the new subproblem adjusts itself adaptively. As a result, an adaptive trust region method is constructed based on the new trust region subproblem. The local and global convergence results of the adaptive trust region method are proved. Numerical results indicate that the new method is very efficient.
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References
Schnabel, R. B., Eskow, E., A new modified Cholesky factorization, SIAM Journal on Scientific Computing, 1990, 11(6): 1136–1158.
Moré, J. J., Recent developments in algorithms and software for trust region methods, in Mathematical Programming: The State of the Art (eds. Bachem, A., Grötchel, M., Korte, B.), Berlin: Springer-Verlag, 1983, 258–287.
Powell, M. J. D., Convergence properties of a class minimization algorithms, in Nonlinear Programming 2 (eds. Mangasarian, O. L., Meyer, R. R., Robinson, S. M.), New York: Academic Press, 1975, 1–27.
Powell, M. J. D., On the global convergence of trust region algorithms for unconstrained optimization, Mathematical Programming, 1984, 29(3): 297–303.
Schultz, G. A., Schnabel, R. B., Byrd, R. H., A family of trust-region-based algorithms for unconstrained minimization with strong global convergence, SIAM Journal on Numerical Analysis, 1985, 22(1): 47–67.
Yuan, Y., On the convergence of trust region algorithms, Mathematica Numerica Sinica (in Chinese), 1996, 16(3): 333–346.
Yuan, Y., Sun, W., Optimization: Theorem and Algorithm (in Chinese), Beijing: Science Press, 1997.
Zhang Xiangsun, Chapter 11: NN models for general nonlinear programming, in Neural Networks in optimization, Dordrecht/Boston/London: Kluwer Academic Publishers, 2000.
Byrd, R. H., Schnabel, R. B., Schultz, G. A., Approximate solution of the trust region problem by minimization over two-dimension subspace, Mathematical Programming, 1988, 40(3): 247–263.
Gay, D. M., Algorithm 611 subroutine for unconstrained minimization using a model/trust region approach, Transaction of the ACM on Mathematical Software, 1983, 9(4): 503–524.
Moré, J. J., Sorensen, D. C., Computing a trust region step, SIAM Journal on Scientific and Statistical Computing, 1983, 4(3): 553–572.
Pham Dinh, T., Hoai An, L T., A D.C. optimization algorithm for solving the trust region subproblem, SIAM Journal on Optimization, 1998, 8(2): 476–505.
Zhang, J. Z., Xu, C., A class of indefinite dogleg path methods for unconstrained minimization, SIAM Journal on Optimization, 1999, 9(3): 646–667.
Nocedal, J., Theory of algorithms for unconstrained optimization, in Acta Numerica 1992 (ed. Iserles, A.), Cambridge: Cambridge University Press, 1992, 199–242.
Sartenaer, A., Automatic determination of an initial trust region in nonlinear programming, SIAM Journal on Scientific Computing, 1997, 18(6): 1788–1803.
Moré, J. J., Garbow, B. S., Hillstrom, K. E., Testing unconstrained optimization software, Transaction of the ACM on Mathematical Software, 1981, 7(1): 17–41.
Nocedal, J., Yuan, Y., Combining trust region and linear search techniques, in Advances in Nonlinear Programming (ed. Yuan, Y.), Dordrecht: Kluwer Academic Publishers, 1996.
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Zhang, X., Zhang, J. & Liao, L. An adaptive trust region method and its convergence. Sci. China Ser. A-Math. 45, 620–631 (2002). https://doi.org/10.1360/02ys9067
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DOI: https://doi.org/10.1360/02ys9067