## L-BFGS-B - Fortran Subroutines for Large-Scale Bound Constrained Optimization (1994)

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Venue: | ACM Trans. Math. Software |

Citations: | 38 - 2 self |

### BibTeX

@TECHREPORT{Zhu94l-bfgs-b-,

author = {Ciyou Zhu and Richard H. Byrd and Peihuang Lu and Jorge Nocedal},

title = {L-BFGS-B - Fortran Subroutines for Large-Scale Bound Constrained Optimization},

institution = {ACM Trans. Math. Software},

year = {1994}

}

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### Abstract

L-BFGS-B is a limited memory algorithm for solving large nonlinear optimization problems subject to simple bounds on the variables. It is intended for problems in which information on the Hessian matrix is difficult to obtain, or for large dense problems. L-BFGS-B can also be used for unconstrained problems, and in this case performs similarly to its predecessor, algorithm L-BFGS (Harwell routine VA15). The algorithm is implemented in Fortran 77. Categories and Subject Descriptors: G.1.6 [Numerical Analysis]: Optimization -- gradient methods; G.4 [Mathematics of Computing]: Mathematical Software. General Terms: Algorithms Additional Key Words and Phrases: variable metric method, large scale optimization, nonlinear optimization, limited memory method. 1 Department of Electrical Engineering and Computer Science, Northwestern University, Evanston Il 60208. These authors were supported by National Science Foundation Grants CCR-9101359 and ASC-9213149, and by Department of Energy Grant DE-...

### Citations

1341 |
Practical Optimization
- Gill, Murray, et al.
- 1981
(Show Context)
Citation Context ...orithm capable of handling bounds on the variables; other published codes [5], [6], [13], [16] are only able to solve unconstrained problems. We note also that the nonlinear conjugate gradient method =-=[14]-=-, which is used for solving many large unconstrained problems, has not been adequately extended to handle bounds on the variables, and L-BFGS-B can be used in its place. The advantages of L-BFGS-B are... |

486 | On the limited memory BFGS method for large scale optimization
- Liu, Nocedal
- 1989
(Show Context)
Citation Context ...h the output of L-BFGS-B and its documentation, make reference to the projected gradient. L-BFGS-B is an extension of the limited memory algorithm (L-BFGS) for unconstrained optimization described in =-=[16]-=- and implemented as Harwell routine VA15 [12]. The main improvement is the ability of L-BFGS-B to deal with bounds on the variables. Even though this requirement makes the new algorithm far more compl... |

316 |
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
- jun, Schnabel
- 1983
(Show Context)
Citation Context ...emory matrices when bounds on the variables are present. In L-BFGS-B the matrix update is skipped when y T k s k \Gammag T k s ksepsmch; where y k = g k+1 \Gamma g k and s k = x k+1 \Gamma x k ; (see =-=[11]-=-). This ensures that y T k s k is sufficiently positive. The user can determine how many times the BFGS update was skipped by printing the variable isave(26). Our numerical experience indicates that s... |

267 | A limited memory algorithm for bound constrained optimization
- Byrd, Lu, et al.
- 1995
(Show Context)
Citation Context ...x of f is not required. For this reason the algorithm can be useful for solving large problems in which the Hessian matrix is difficult to compute or is dense. The algorithm is described in detail in =-=[8]-=-, and proceeds roughly as follows. At each iteration a limited memory BFGS approximation to the Hessian is updated. This limited memory matrix is used to define a quadratic model of the objective func... |

214 | Updating quasi-newton matrices with limited storage - Nocedal - 1980 |

151 | Cute: Constrained and Unconstrained Testing Environment
- BONGARTZ, CONN, et al.
- 1995
(Show Context)
Citation Context ... driver shows how to generate all desired output in this case. driver4.f is an extensive driver. It runs the code on 67 test problems (33 bound constrained, 34 unconstrained) from the CUTE collection =-=[4]-=-, each with three different subspace minimization methods. This driver is used for testing and profiling the code. 4. Termination and Error Messages. The user can terminate execution at various stages... |

103 | Representations of quasi-Newton matrices and their use in limited memory methods
- Byrd, Nocedal, et al.
- 1994
(Show Context)
Citation Context ...line search is performed along the search direction using the subroutine described in [17]. A novel feature of the algorithm is that the limited memory BFGS matrices are represented in a compact form =-=[7]-=- that is efficient for bound constrained problems. The user can control the amount of storage required by L-BFGS-B by selecting a parameter m that determines the number of BFGS corrections saved. The ... |

87 |
LANCELOT: A Fortran package for Large-scale Nonlinear Optimization (Release
- Conn, Gould, et al.
- 1992
(Show Context)
Citation Context ...tested only bound constrained problems with ns5 and unconstrained problems with ns100. As a benchmark we also present the results obtained by the SR1 and Exact Hessian options of the LANCELOT package =-=[10]-=-. LANCELOT was run using all its default options. All runs were performed on a Sparcstation-2 with 32Mb of main memory, and all runs were terminated when the norm of the projected gradient is less tha... |

84 | Line search algorithms with guaranteed sufficient decrease
- MorĂ©, Thuente
- 1994
(Show Context)
Citation Context ...h direction is defined to be the vector leading from the current iterate to this approximate minimizer. Finally a line search is performed along the search direction using the subroutine described in =-=[17]-=-. A novel feature of the algorithm is that the limited memory BFGS matrices are represented in a compact form [7] that is efficient for bound constrained problems. The user can control the amount of s... |

81 |
Projected Newton methods for optimization problems with simple constraints
- Bertsekas
- 1982
(Show Context)
Citation Context ...his limited memory matrix is used to define a quadratic model of the objective function f . A search direction is then computed using a two-stage approach: first, the gradient projection method [15], =-=[3]-=-, [18],[9] is used to identify a set of active variables, i.e. variables that will be held at their bounds; then the quadratic model is approximately minimized with respect to the free variables. The ... |

48 |
Some numerical experiments with variable storage quasinewton algorithms, Mathematical programming 45
- Gilbert, Lemarechal
- 1989
(Show Context)
Citation Context ...ns: L-BFGS-B requires 8 more n-vectors of storage. L-BFGS-B is, at present, the only limited memory quasi-Newton algorithm capable of handling bounds on the variables; other published codes [5], [6], =-=[13]-=-, [16] are only able to solve unconstrained problems. We note also that the nonlinear conjugate gradient method [14], which is used for solving many large unconstrained problems, has not been adequate... |

37 |
Algorithms for bound constrained quadratic programming problems
- Mor'e, Toraldo
- 1989
(Show Context)
Citation Context ...imited memory matrix is used to define a quadratic model of the objective function f . A search direction is then computed using a two-stage approach: first, the gradient projection method [15], [3], =-=[18]-=-,[9] is used to identify a set of active variables, i.e. variables that will be held at their bounds; then the quadratic model is approximately minimized with respect to the free variables. The search... |

25 |
Constrained minimization problems
- LEVITIN, POLJAK
- 1966
(Show Context)
Citation Context ...ted. This limited memory matrix is used to define a quadratic model of the objective function f . A search direction is then computed using a two-stage approach: first, the gradient projection method =-=[15]-=-, [3], [18],[9] is used to identify a set of active variables, i.e. variables that will be held at their bounds; then the quadratic model is approximately minimized with respect to the free variables.... |

17 |
Testing a class of methods for solving minimization problems with simple bounds on the variables
- Toint
- 1988
(Show Context)
Citation Context ...d memory matrix is used to define a quadratic model of the objective function f . A search direction is then computed using a two-stage approach: first, the gradient projection method [15], [3], [18],=-=[9]-=- is used to identify a set of active variables, i.e. variables that will be held at their bounds; then the quadratic model is approximately minimized with respect to the free variables. The search dir... |

7 |
BBVSCG {A variable storage algorithm for function minimization
- Buckley, LeNir
- 1985
(Show Context)
Citation Context ...applications: L-BFGS-B requires 8 more n-vectors of storage. L-BFGS-B is, at present, the only limited memory quasi-Newton algorithm capable of handling bounds on the variables; other published codes =-=[5]-=-, [6], [13], [16] are only able to solve unconstrained problems. We note also that the nonlinear conjugate gradient method [14], which is used for solving many large unconstrained problems, has not be... |

6 | LANCELOT: aFORTRAN package for large-scale nonlinear optimization (Release A)", Number 17 - Conn, Gould, et al. - 1992 |

5 | Private communication - Averick, More - 1992 |

3 |
Remark on algorithm 630
- Buckley
- 1989
(Show Context)
Citation Context ...cations: L-BFGS-B requires 8 more n-vectors of storage. L-BFGS-B is, at present, the only limited memory quasi-Newton algorithm capable of handling bounds on the variables; other published codes [5], =-=[6]-=-, [13], [16] are only able to solve unconstrained problems. We note also that the nonlinear conjugate gradient method [14], which is used for solving many large unconstrained problems, has not been ad... |

2 |
Release 10
- Library
- 1990
(Show Context)
Citation Context ...n, make reference to the projected gradient. L-BFGS-B is an extension of the limited memory algorithm (L-BFGS) for unconstrained optimization described in [16] and implemented as Harwell routine VA15 =-=[12]-=-. The main improvement is the ability of L-BFGS-B to deal with bounds on the variables. Even though this requirement makes the new algorithm far more complex than its predecessor, the two codes perfor... |

1 |
The MINPACK-2 package", in preparation
- Averick, More
(Show Context)
Citation Context ...scribed occur only when rounding errors begin to dominate the computation. Machine and Scale Dependencies. L-BFGS-B computes the machine precision epsmch by means of the routine dpmeps from MINPACK-2 =-=[2]-=-. The machine precision epsmch is used only twice in the algorithm: in the stopping test (1) and in the skipping criterion for BFGS updating described below. These two computations are therefore machi... |