A compact limited memory method for large scale unconstrained optimization. (English) Zbl 1114.90072

Summary: A compact limited memory method for solving large scale unconstrained optimization problems is proposed. The compact representation of the quasi-Newton updating matrix is derived to the use in the form of limited memory update in which the vector \(y_{k}\) is replaced by a modified vector \(\hat y_k\) so that more available information about the function can be employed to increase the accuracy of Hessian approximations. The global convergence of the proposed method is proved. Numerical tests on commonly used large scale test problems indicate that the proposed compact limited memory method is competitive and efficient.


90C06 Large-scale problems in mathematical programming
90C52 Methods of reduced gradient type
90C30 Nonlinear programming


minpack; L-BFGS
Full Text: DOI


[1] R.H. Byrd, J. Nocedal, R.B. Schnabel, Representations of quasi-Newton matrices and their use in limited memory methods, Technical Report NAM-03, Department of Electrical Engineering and Computer Science, Northwestern University, 1992. · Zbl 0809.90116
[2] Hande Yurttan Benson, Cuter models. Available from: <http://www.orfe.princeton.edu/rvdb/ampl/nlmodels/cute/>.
[3] Liu, D.C.; Nocedal, J., On the limited memory BFGS method for large scale optimization, Mathematical programming, 45, 503-528, (1989) · Zbl 0696.90048
[4] W.D. Lin, C.X. Xu, Global convergence properties of convex Broyden quasi-Newton methods based on the new quasi-Newton equations, in: Presented at the International Conference on Nonlinear Programming and Variational Inequalities, City University of Hong Kong, December, 1998.
[5] More, J.J.; Garbow, B.S.; Hillstrom, K.E., Testing unconstrained optimization software, ACM transactions on mathematical software, 7, 17-41, (1981) · Zbl 0454.65049
[6] Morales, J.L.; Nocedal, J., Automatic preconditioning by limited memory quasi-Newton updating, SIAM journal on optimization, 10, 1079-1096, (1996) · Zbl 1020.65019
[7] Nazareth, L., A relationship between BFGS and conjugate gradient algorithms and its implications for new algorithms, SIAM journal on numerical analysis, 16, 794-800, (1979) · Zbl 0424.65030
[8] Nocedal, J., Updating quasi-Newton matrices with limited storage, Mathematics of computation, 35, 773-782, (1980) · Zbl 0464.65037
[9] Schnabel, R.B.; Chow, T., Tensor methods for unconstrained optimization using second derivatives, SIAM journal on optimization, 1, 293-315, (1991) · Zbl 0758.65047
[10] Toint, Ph.L., Some numerical results using a sparse matrix updating formula in unconstrained optimization, Mathematics of computation, 32, 839-851, (1978) · Zbl 0381.65036
[11] Ph.L. Toint, Test problems for partially separable optimization and results for the routine PSPMIN, Report Nn8314, Department of Mathematics, Faculte’s Universitaries de Namur, Namur, Belgium, 1983.
[12] Xu, Ch.X.; Chen, Z.P.; Li, N.C., Advanced optimization algorithms, (2002), Scientific Publisher Beijing, (in Chinese)
[13] Xu, Ch.X.; Zhang, J.Z., A survey of quasi-Newton equations and quasi-Newton methods for optimization, Annals of operations research, 103, 213-234, (2001) · Zbl 1007.90069
[14] Yuan, Y.; Byrd, R., Non-quasi-Newton updates for unconstrained optimization, Journal of computation mathematics, 13, 95-107, (1995) · Zbl 0823.65062
[15] Zhang, J.Z.; Deng, N.Y.; Chen, L.H., New quasi-Newton equation and related methods for unconstrained optimization, Journal of optimization theory and application, 102, 147-167, (1999) · Zbl 0991.90135
[16] Zhang, J.Z.; Xu, Ch.X., Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations, Journal of computational and applied mathematics, 137, 269-278, (2001) · Zbl 1001.65065
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