## A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems (1999)

Venue: | SIAM JOURNAL ON SCIENTIFIC COMPUTING |

Citations: | 46 - 1 self |

### BibTeX

@TECHREPORT{Branch99asubspace,,

author = {Mary Ann Branch and Thomas F. Coleman and Yuying Li},

title = {A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems},

institution = {SIAM JOURNAL ON SCIENTIFIC COMPUTING},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

A subspace adaptation of the Coleman-Li trust region and interior method is proposed for solving large-scale bound-constrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full-space version. Computational

### Citations

1544 |
Practical Optimization
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- 1981
(Show Context)
Citation Context ...t guaranteed to generate sufficient negative curvature; nonetheless, as indicated in [7], the MPCG output will satisfy the angle property. Finally we consider a modified Cholesky factorization, e.g., =-=[17]-=-, to obtain a negative curvature vector. Lemma 5. Assume that { ˆ Mk} is indefinite or negative definite and {dk} is obtained from the modified Cholesky method. Then the sequence {dk = D −1 k ˆ dk} ha... |

234 |
Computing a trust region step
- Moré, Sorensen
- 1983
(Show Context)
Citation Context ...or unconstrained minimization: (1.2) min s∈ℜ n{ψk(s) : ‖Dks‖2 ≤∆k}, where ψk(s) is a quadratic function, Dk is a positive diagonal matrix, and ∆k is a positive scalar. The method of Moré and Sorensen =-=[4]-=- can be directly applied to (1.2) if sparse Cholesky factorizations can be computed efficiently. However, this method is unsuitable for large-scale problems if the Hessian matrix is not explicitly ava... |

224 |
An interior trust region approach for nonlinear minimization subject to bounds
- Coleman, Li
- 1996
(Show Context)
Citation Context ..., inexact Newton step, conjugate gradients, bound-constrained problem, box constraints AMS subject classifications. 65K05, 90C06, 90C30 PII. S1064827595289108 1. Introduction. Recently Coleman and Li =-=[1, 2, 3]-=- proposed two interior and reflective Newton methods to solve the bound-constrained minimization problem, i.e., min x∈ℜn{f(x) : l≤x≤u}, (1.1) where l ∈ {ℜ ∪ {−∞}} n ,u∈ {ℜ ∪ {∞}} n ,l<u,and f is a smo... |

209 |
Lanczos algorithms for large symmetric eigenvalue computations. Volumes 1 and 2
- Cullum, Willoughby
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Citation Context ...negative curvature vector sequence with the angle property, orthogonality needs to be maintained in the Lanczos process. In practice maintaining orthogonality can be a delicate and expensive business =-=[16]-=-. A second (and cheaper) strategy is to employ a modified preconditioned conjugate gradient scheme, e.g., modified preconditioned conjugate gradient algorithm (MPCG) in Fig. 4. Unfortunately, this pro... |

161 | CUTE: Constrained and unconstrained testing environment
- Bongartz, Conn, et al.
- 1995
(Show Context)
Citation Context ...ubspace trust region approach than in the Steihaug trust region method, which SBMIN employs. This is consistent with the remarks in section 3 and [7, 8]. 2BIGGSB2 is a modification of problem BIGGSB1 =-=[20]-=-. Active set methods, such as LANCELOT, are likely to struggle with the identification of the correct active set on this modified version (from the given starting point). BIGGSB2: f(x) =(x1−1) 2 +(1−x... |

140 |
The conjugate gradient method and trust regions in large scale optimization
- Steihaug
- 1983
(Show Context)
Citation Context ... trust region algorithm, is yet to be investigated. We take the view that solving the full-space trust region subproblem (1.2) is too costly for a large-scale problem. This view is shared by Steihaug =-=[6]-=-, who proposes an approximate (conjugate gradient) approach. Steihaug’s approach to (1.2) is viable, although the computational studies in [7, 8] indicate that important negative curvature information... |

106 | LANCELOT: a Fortran package for Large-scale Nonlinear Optimization - Toint - 1992 |

92 |
LANCELOT: A Fortran Package for LargeScale Nonlinear Optimization (Release
- Conn, Gould, et al.
- 1992
(Show Context)
Citation Context ...space trust region method is a promising way to solve large-scale bound-constrained nonlinear minimization problems. Finally, our subspace method is competitive with the active set method in LANCELOT =-=[10]-=- and we include results on test problems with negative curvature where our subspace method outperforms the LANCELOT method. The paper is organized as follows. In section 2 we briefly summarize the exi... |

91 |
Iterative solution of problems of linear and quadratic programming
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- 1967
(Show Context)
Citation Context ... demonstrate the benefits of our affine scaling, reflection, and subspace techniques with computational results. First, for (1.1), our affine scaling technique outperforms the classical Dikin scaling =-=[9]-=-, at least in the context of our algorithm. Second, we examine our method with and without reflection. We show that the reflection technique can substantially reduce the number of minimization iterati... |

84 | On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds
- Coleman, Li
- 1994
(Show Context)
Citation Context ...AS.3) and (AS.4) is not difficult. For example, one can choose sk so that ψk(sk) is the minimum of the values ψ∗ k [pk] and ψ∗ k [−D−2 k gk]. However, this does not lead to an efficient algorithm. In =-=[3, 2]-=-, we utilized a reflection technique to permit further possible reduction of the objective function along a reflection path on the boundary. We found in [3, 2] that this reflection process significant... |

56 | A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables
- COLEMAN, LI
- 1996
(Show Context)
Citation Context ..., inexact Newton step, conjugate gradients, bound-constrained problem, box constraints AMS subject classifications. 65K05, 90C06, 90C30 PII. S1064827595289108 1. Introduction. Recently Coleman and Li =-=[1, 2, 3]-=- proposed two interior and reflective Newton methods to solve the bound-constrained minimization problem, i.e., min x∈ℜn{f(x) : l≤x≤u}, (1.1) where l ∈ {ℜ ∪ {−∞}} n ,u∈ {ℜ ∪ {∞}} n ,l<u,and f is a smo... |

46 |
A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties
- Shultz, Schnabel, et al.
- 1985
(Show Context)
Citation Context ...n in unconstrained minimization. We discuss two approaches that have been shown effective in approximating a full-space trust region solution in unconstrained minimization. Shultz, Schnabel, and Byrd =-=[13]-=- suggest replacing the full trust region subproblem in the unconstrained setting by (3.1) min s∈ℜn{ψk(s) : ‖s‖2≤∆k,s∈Sk}, where Sk is a low-dimensional subspace. Another possible consideration for the... |

42 |
Testing a class of methods for solving minimization problems with simple bounds on the variables
- CONN, GOULD, et al.
- 1988
(Show Context)
Citation Context ...ort the results of the STIR method using the modified Cholesky factorization. Table 3 lists the number of major iterations required for some standard test problems (for details of these problems, see =-=[19]-=-). (For all the results in this paper, the number of iterations is the same as the number of objective function evaluations.) The problem sizes vary from 100 to 10, 000. The results in Table 3 indicat... |

39 |
Approximate solution of the trust region problem by minimization over two-dimensional subspaces
- Byrd, Schnabel, et al.
- 1988
(Show Context)
Citation Context ..., the following conditions must be met. First, when the matrix ˆ Mk is indefinite or negative definite, a “sufficient negative curvature” condition must be carried over from the unconstrained setting =-=[14]-=-. To this end, we can require that sufficient negative curvature of the matrix ˆ Mk be captured if ˆ Mk is indefinite or negative definite; i.e., Sk must contain a vector wk = D −1 k ˆwk such that ˆw ... |

27 |
The molecule problem: Determining conformation from pairwise distances. Cornell University
- Hendrickson
- 1991
(Show Context)
Citation Context ...Before we investigate a subspace adaptation of TIR, we demonstrate the effectiveness of our reflection idea and affine scaling technique. We consider random problem instances of molecule minimization =-=[11, 12]-=-, which minimize a quartic subject to bounds on the variables. Tables 1 and 2 list the average number of iterations (more than 10 random test problem instances for each entry) required for the differe... |

21 |
Minimization of a large scale quadratic function subject to an ellipsoidal constraint
- Sorensen
- 1994
(Show Context)
Citation Context ...ed efficiently. However, this method is unsuitable for large-scale problems if the Hessian matrix is not explicitly available or (sparse) Cholesky factorizations are too expensive. Recently, Sorensen =-=[5]-=- proposed a new method for solving the subproblem (1.2) using matrix-vector multiplications. Nonetheless, the effectiveness of the Sorensen approach for large-scale minimization, particularly in the c... |

19 | Testing a class of methods for solving minimization problems with simple bounds on the variables - Toint - 1988 |

13 |
Large-scale numerical optimization: Introduction and overview
- Coleman
- 1993
(Show Context)
Citation Context ...Before we investigate a subspace adaptation of TIR, we demonstrate the effectiveness of our reflection idea and affine scaling technique. We consider random problem instances of molecule minimization =-=[11, 12]-=-, which minimize a quartic subject to bounds on the variables. Tables 1 and 2 list the average number of iterations (more than 10 random test problem instances for each entry) required for the differe... |

2 |
Inexact Reflective Newton Methods for Large-Scale Optimization Subject to Bound Constraints
- Branch
- 1996
(Show Context)
Citation Context ...or a large-scale problem. This view is shared by Steihaug [6], who proposes an approximate (conjugate gradient) approach. Steihaug’s approach to (1.2) is viable, although the computational studies in =-=[7, 8]-=- indicate that important negative curvature information can be missed, causing a significant increase in the number of minimization iterations. In this paper, we propose an alternative: an approximate... |

1 | Testing unconstrained optimization software - Hillstrom - 1981 |

1 |
A Computational Study of Two Approximations to the Trust-Region Subproblem
- Branch, Coleman, et al.
- 1998
(Show Context)
Citation Context ...or a large-scale problem. This view is shared by Steihaug [6], who proposes an approximate (conjugate gradient) approach. Steihaug’s approach to (1.2) is viable, although the computational studies in =-=[7, 8]-=- indicate that important negative curvature information can be missed, causing a significant increase in the number of minimization iterations. In this paper, we propose an alternative: an approximate... |

1 | Inexact Re¯ective Newton Methods for Large-Scale Optimization Subject to Bound Constraints - Branch - 1995 |