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A new subspace limited memory BFGS algorithm for large-scale bound constrained optimization. (English) Zbl 1114.65069
A new algorithm that combines an active set strategy with the gradient projection method is presented. As in the work by F. Facchinei, S. Lucidi and L. Palagi [SIAM J. Opt., 12, 1100–1125 (2002; Zbl 1035.90103)] the authors avoid the necessity of finding an exact minimizer of a quadratic subproblem with bound constraints. The algorithm has the following properties: All iterates are feasible and the sequence of the objective function values is decreasing; rapid changes in the active set are allowed; a global convergence theory is established.
Moreover, it reserves the advantage of the effective active set identified technique by Facchinei, Lucidi and Palagi [loc. cit.] and uses the superiority of the subspace limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [see Q. Ni and Y. X. Yuan, Math. Comp., 66, 1509–1520 (1997; Zbl 0886.65065)] which has been proved much suit for solving large-scale problems. Namely, the active sets are based on a guessing technique to be identified at each iteration, the search direction in the free subspace is determined by a limited memory BFGS algorithm, which provides an efficient means for attacking large-scale optimizatuin problems. The implementations of the method on CUTE test problems are described.

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C06 Large-scale problems in mathematical programming
Full Text: DOI
[1] Bertsekas, D.P., Projected Newton methods for optimization problems with simple constrains, SIAM J. contr. opt., 20, 221-246, (1982) · Zbl 0507.49018
[2] Birgin, E.G.; Martínez, J.M., Large−scale active-set box-constrained optimization method with spectral projected gradients, Comput. opt. appl., 23, 101-125, (2002) · Zbl 1031.90012
[3] Burke, J.V.; Moré, J.J., On the identification of active constraints, SIAM J. numer. anal., 25, 1197-1211, (1988) · Zbl 0662.65052
[4] Burke, J.V.; Moré, J.J., Exposing constrints, SIAM J. opt., 4, 573-595, (1994) · Zbl 0809.65058
[5] Ryrd, R.H.; Lu, P.H.; Nocedal, J., A limited memory algorithm for bound constrained optimization, SIAM J. statist. sci. comput., 16, 1190-1208, (1995) · Zbl 0836.65080
[6] Ryrd, R.H.; Nocedal, J.; Schnabel, R.B., Representations of quasi-Newton matrices and their use in limited memory methods, Math. program., 63, 129-156, (1994) · Zbl 0809.90116
[7] Calamai, P.; Moré, J.J., Projected gradient for linearly constrained programs, Math. program., 39, 93-116, (1987) · Zbl 0634.90064
[8] Ccoleman, T.F.; Li, Y., An interior trust region approach for nonlinear minimization subject to bounds, SIAM J. opt., 6, 418-445, (1996) · Zbl 0855.65063
[9] Conn, A.R.; Gould, N.I.M.; Toint, Ph. L., Global convergence of a class of trust region algorithm for optimization with simple bounds, SIAM J. numer. anal., 25, 433-460, (1988) · Zbl 0643.65031
[10] Conn, A.R.; Gould, N.I.M.; Toint, Ph. L., : constrained and unconstrained testing environment, ACM trans. math. softw., 21, 123-160, (1995) · Zbl 0886.65058
[11] Conn, A.R.; Gould, N.I.M.; Toint, Ph. L., : A Fortran package for large-scale nonlinear optimization (release A), Springer series in computational mathematics, (1992), Springer Verlag Heidelberg, Berlin, New York
[12] Facchinei, F.; Júdice, J.; Soares, J., An active set Newton algorithm for large-scale nonlinear programs with box canstranits, SIAM J. opt., 8, 158-186, (1998) · Zbl 0911.90301
[13] Facchinei, F.; Lucidi, S.; Palagi, L., A truncated Newton algorithm for large scale box constrained optimization, SIAM J. opt., 12, 1100-1125, (2002) · Zbl 1035.90103
[14] Gould, N.I.M.; Orban, D.; Toint, Ph.L., Numerical methods for large-scale nonlinear optimization, Acta numer., 14, 299-361, (2005) · Zbl 1119.65337
[15] Goldfarb, D., Extension of davidon’s variable metric algorithm to maximization under linear inequality and constraints, SIAM J. appl. math., 17, 739-764, (1969) · Zbl 0185.42602
[16] C.T. Kelley, Iterative methods for optimization, Philadelphia, PA, 1999. · Zbl 0934.90082
[17] Lin, C.J.; Moré, J.J., Nowton’s method for large bound-constrained optimization problems, SIAM J. opt., 9, 1100-1127, (1999) · Zbl 0957.65064
[18] Lescrenier, M., Convergence of trust region algorithm for optimization with bounds when strict complementarity does not hold, SIAM J. numer. anal., 28, 467-695, (1991) · Zbl 0726.65068
[19] Luenberger, D.G., Introduction to linear and nonlinear programming, (1973), Addison-Wesley Reading, MA, Ch. 11 · Zbl 0241.90052
[20] Moré, J.J.; Toraldo, G., On the solution of large quadratic programming problems with bound constraints, SIAM J. opt., 1, 93-113, (1991) · Zbl 0752.90053
[21] B.A. Murtagh, M.A. Saunders, User’s Guide, Report SOL 83-20R, Systems Optimization Laboratory, Stanford University (revised July 1998).
[22] Ni, Q., A subspace projected conjuagte gradient algorithm for large bound constrained quadratic programming, Numer. math. (a journal of Chinese universities), 7, 51-60, (1998) · Zbl 0909.65036
[23] Ni, Q.; Yuan, Y.X., A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization, Math. comp., 66, 1509-1520, (1997) · Zbl 0886.65065
[24] Powell, M.J.D., A fast algorithm for nonlinearly constrained optimization calculations, Numer. anal., 155-157, (1978) · Zbl 0374.65032
[25] R.J. Vanderbei, : An interior point code for quadratic programming, Technical Report, Statistics and Operation Research, Princeton University, SOR-94-15, 1994.
[26] Zhu, C.Y.; Byrd, R.H.; Lu, P.H.; Nocedal, J., Fortran subroutines for large-scale bound constrained optimization, ACM trans. math. softw., 23, 550-560, (1997) · Zbl 0912.65057
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