A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization.

*(English)*Zbl 0886.65065The following nonlinear programming problem with simple bounds on variables
\[
\text{minimize }f(x)\quad\text{subject to }\ell\leq x\leq u
\]
is considered. The objective function \(f(x)\) is assumed to be twice continuously differentiable, \(\ell\) and \(u\) are given bound vectors in \(\mathbb{R}^n\), and \(n\) is the number of variables, which is assumed to be large.

The given subspace limited memory quasi-Newton algorithm does not need to solve any subproblems. The search direction of the algorithm consists of three parts: a subspace quasi-Newton direction, and two subspace gradient and modified gradient directions. The global convergence of the method is proved and some numerical results are given.

The given subspace limited memory quasi-Newton algorithm does not need to solve any subproblems. The search direction of the algorithm consists of three parts: a subspace quasi-Newton direction, and two subspace gradient and modified gradient directions. The global convergence of the method is proved and some numerical results are given.

Reviewer: H.Benker (Merseburg)

##### MSC:

65K05 | Numerical mathematical programming methods |

90C06 | Large-scale problems in mathematical programming |

90C30 | Nonlinear programming |

##### Keywords:

subspace quasi-Newton method; limited memory; projected search; large scale problem; bound constrained optimization; nonlinear programming; quasi-Newton algorithm; convergence; numerical results
PDF
BibTeX
XML
Cite

\textit{Q. Ni} and \textit{Y. Yuan}, Math. Comput. 66, No. 220, 1509--1520 (1997; Zbl 0886.65065)

Full Text:
DOI

##### References:

[1] | R.H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, Report NAM-8, EECS Department, Northwestern University, 1994. · Zbl 0836.65080 |

[2] | Richard H. Byrd, Jorge Nocedal, and Robert B. Schnabel, Representations of quasi-Newton matrices and their use in limited memory methods, Math. Programming 63 (1994), no. 2, Ser. A, 129 – 156. · Zbl 0809.90116 · doi:10.1007/BF01582063 · doi.org |

[3] | I. Bongartz, A.R. Conn, N. Gould, and Ph.L. Toint, CUTE: constrained and unconstrained testing environment, Research Report, IBM T.J. Watson Research Center, Yorktown, USA. · Zbl 0886.65058 |

[4] | A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Global convergence of a class of trust region algorithms for optimization with simple bounds, SIAM J. Numer. Anal. 25 (1988), no. 2, 433 – 460. · Zbl 0643.65031 · doi:10.1137/0725029 · doi.org |

[5] | Andrew R. Conn, Nicholas I. M. Gould, and Philippe L. Toint, Testing a class of methods for solving minimization problems with simple bounds on the variables, Math. Comp. 50 (1988), no. 182, 399 – 430. · Zbl 0645.65033 |

[6] | A.R. Conn, N.I.M. Gould and Ph.L. Toint, LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A), Number 17 in Springer Series in Computational Mathematics, Springer Verlag, Heidelberg, New York, 1992. CMP 93:12 · Zbl 0761.90087 |

[7] | R. Fletcher, Practical methods of optimization, 2nd ed., A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. · Zbl 0905.65002 |

[8] | P. Lu, Bound constrained nonlinear optimization and limited memory methods, Ph.D. Thesis, Dept. of Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois, 1992. |

[9] | Jorge J. Moré and Gerardo Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optim. 1 (1991), no. 1, 93 – 113. · Zbl 0752.90053 · doi:10.1137/0801008 · doi.org |

[10] | Qin Ni, General large-scale nonlinear programming using sequential quadratic programming methods, Bayreuth. Math. Schr. 45 (1993), 133 – 236 (English, with German summary). Dissertation, Universität Bayreuth, Bayreuth, 1993. · Zbl 0807.90105 |

[11] | Jorge Nocedal, Updating quasi-Newton matrices with limited storage, Math. Comp. 35 (1980), no. 151, 773 – 782. · Zbl 0464.65037 |

[12] | M. J. D. Powell, A fast algorithm for nonlinearly constrained optimization calculations, Numerical analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), Springer, Berlin, 1978, pp. 144 – 157. Lecture Notes in Math., Vol. 630. |

[13] | C. Zhu, R.H. Byrd, P. Lu, and J. Nocedal, L-BFGS-B Fortran subroutines for large-scale bound constrained optimization, Report NAM-11, EECS Department, Northwestern University, 1994. · Zbl 0912.65057 |

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.