### Citations

3305 | Numerical optimization
- Nocedal, Wright
- 2000
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Citation Context ... AND ROUMMEL F. MARCIA One of the advantages of using an L-BFGS quasi-Newton is that there is an efficient recursion relation to compute products with B−1m . Given a vector z, the following algorithm =-=[20, 21]-=- terminates with r △= B−1m z: Algorithm 1: Two-loop recursion to compute r = B−1m z. q ← z; for k = m− 1, . . . , 0 ρk ← 1/(yTk sk); αk ← ρksTk q; q ← q − αkyk; end r ← B−10 q; for k = 0, . . . , m− 1... |

778 | On the limited memory BFGS method for large scale optimization
- Liu, Nocedal
- 1989
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Citation Context ... aiai T + m−1∑ i=0 bib T i , (4) where ai = Bisi√ sTi Bisi , bi = yi√ yTi si , B0 = γ −1 m I, (5) and γm > 0 is a constant. In practice, γm is often defined to be γm △ = sTm−1ym−1/‖ym−1‖2 (see, e.g., =-=[17]-=- or [20]). In order to maintain that the sequence {Bi} is positive definite for i = 1, . . .m, each of the accepted pairs must satisfy yTi si > 0 for i = 0, . . . , m− 1. 4 JENNIFER B. ERWAY AND ROUMM... |

478 |
Numerical Methods for Unconstrained Optimization and Nonlinear Equations
- Schnabel
- 1983
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Citation Context ...with the LBFGS quasi-Newton matrix, which requires O(Mn) multiplications (see, e.g., [21]). Further details on L-BFGS updates can found in [21]; further background on the BFGS updates can be found in =-=[7]-=-. Without loss of generality, for the duration of the paper we assume that B is a symmetric positive-definite quasi-Newton matrix formed using m (m ≤M) L-BFGS updates. 3. The Moré-Sorensen Sequential... |

375 | Benchmarking optimization software with performance profiles
- Dolan, Moré
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Citation Context ...vals 4359 4974 Total time (subproblem) 1.71e+01 1.68e+01 12 JENNIFER B. ERWAY AND ROUMMEL F. MARCIA The results of Tables I and II are also summarized using a performance profile (see Dolan and Moré =-=[9]-=-). Let card(S) denote the number of elements in a finite set S. Let P denote the set of problems used for a given numerical experiment. For each method s we define the function πs : [0, rM ] 7→ ℜ+ suc... |

367 |
Updating quasi-Newton matrices with limited storage
- Nocedal
- 1980
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Citation Context ...ure the quasi-Newton matrices remain sufficiently positive definite (see, e.g., [4]). 2. Background In this section, we begin with an overview of the L-BFGS quasi-Newton matrices described by Nocedal =-=[20]-=-, defining notation that will be used throughout the paper. The L-BFGS quasi-Newton method generates a sequence of positive-definite matrices {Bj} from a sequence of vectors {yj} and {sj} defined as y... |

305 |
Computing a Trust Region Step
- Moré, Sorensen
- 1983
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Citation Context ...accuracy. Methods to solve the trust-region subproblem to high accuracy are often based on optimality conditions given in the following theorem (see, e.g., Gay [14], Sorensen [24], Moré and Sorensen =-=[19]-=- or Conn, Gould and Toint [6]): Theorem 1. Let δ be a positive constant. A vector p∗ is a global solution of the trust-region subproblem (1) if and only if ‖p∗‖2 ≤ δ and there exists a unique σ∗ ≥ 0 s... |

187 | CUTE: Constrained and unconstrained testing environment
- Bongartz, Conn, et al.
- 1995
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Citation Context ...s [13, 12] and is able to compute solutions to any user-defined accuracy. This MATLAB implementation is a matrix-free iterative method for large-scale optimization. Numerical experiments on the CUTEr =-=[3, 16]-=-) suggest that using the MSS method as a trust-region subproblem solver can require significantly fewer function and gradient evaluations needed by a trust-region method as compared with the Steihaug-... |

159 | Representations of quasi-newton matrices and their use in limited methods
- Byrd, Nocedal, et al.
- 1994
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Citation Context ...O(M2n). Operations with C0 = B0 + σI and C1 can be easily computed with minimal extra expense since C−10 is a diagonal matrix. It is generally known that M may be kept small (for example, Byrd et al. =-=[5]-=- suggest M ∈ [3, 7]). When M2 ≪ n, the extra storage requirements and computations are affordable. 3.1. Handling small σ. ε In this section we discuss the case of solving (B + σI)p = −g for small σ. F... |

128 | Newton's method with a model trust region modification
- Sorensen
- 1982
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Citation Context ... (1) to any user-defined accuracy. Methods to solve the trust-region subproblem to high accuracy are often based on optimality conditions given in the following theorem (see, e.g., Gay [14], Sorensen =-=[24]-=-, Moré and Sorensen [19] or Conn, Gould and Toint [6]): Theorem 1. Let δ be a positive constant. A vector p∗ is a global solution of the trust-region subproblem (1) if and only if ‖p∗‖2 ≤ δ and there... |

99 |
Computing optimal locally constrained steps
- Gay
- 1981
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Citation Context ...orithm to solve (1) to any user-defined accuracy. Methods to solve the trust-region subproblem to high accuracy are often based on optimality conditions given in the following theorem (see, e.g., Gay =-=[14]-=-, Sorensen [24], Moré and Sorensen [19] or Conn, Gould and Toint [6]): Theorem 1. Let δ be a positive constant. A vector p∗ is a global solution of the trust-region subproblem (1) if and only if ‖p∗‖... |

84 | Ph.L.: CUTEr and SifDec: A constrained and unconstrained testing environment, revisited.
- Gould, Orban, et al.
- 2003
(Show Context)
Citation Context ...s [13, 12] and is able to compute solutions to any user-defined accuracy. This MATLAB implementation is a matrix-free iterative method for large-scale optimization. Numerical experiments on the CUTEr =-=[3, 16]-=-) suggest that using the MSS method as a trust-region subproblem solver can require significantly fewer function and gradient evaluations needed by a trust-region method as compared with the Steihaug-... |

67 |
A hybrid method for nonlinear equations
- Powell
- 1970
(Show Context)
Citation Context ...tion and gradient evaluations are of particular interest when function (or gradient) evaluations are time-consuming, e.g., simulation-based optimization. Some solvers such as the “dogleg” method (see =-=[23, 22]-=-) or the “double-dogleg” strategy (see [8]) Date: May 5, 2014. Key words and phrases. Large-scale unconstrained optimization, trust-region methods, limited-memory quasi-Newton methods, L-BFGS. J. B. E... |

52 |
A Fortran Subroutine for Solving Systems of Nonliner Algebraic Equations
- Powell
- 1970
(Show Context)
Citation Context ...tion and gradient evaluations are of particular interest when function (or gradient) evaluations are time-consuming, e.g., simulation-based optimization. Some solvers such as the “dogleg” method (see =-=[23, 22]-=-) or the “double-dogleg” strategy (see [8]) Date: May 5, 2014. Key words and phrases. Large-scale unconstrained optimization, trust-region methods, limited-memory quasi-Newton methods, L-BFGS. J. B. E... |

48 | Solving the trust-region subproblem using the Lanczos method
- Gould, Lucidi, et al.
- 1999
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Citation Context .... In particular, the Steihaug-Toint method computes an approximate solution to (1) that is guaranteed to achieve at least half the optimal reduction in the quadratic function when the model is convex =-=[25, 15]-=-, but does not specifically seek to solve the minimization problem to high accuracy when the solution lies on the boundary. This paper presents an algorithm to solve (1) to any user-defined accuracy. ... |

16 |
Two new unconstrained optimization algorithms which use function and gradient values
- Mei
- 1979
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Citation Context ... interest when function (or gradient) evaluations are time-consuming, e.g., simulation-based optimization. Some solvers such as the “dogleg” method (see [23, 22]) or the “double-dogleg” strategy (see =-=[8]-=-) Date: May 5, 2014. Key words and phrases. Large-scale unconstrained optimization, trust-region methods, limited-memory quasi-Newton methods, L-BFGS. J. B. Erway is supported in part by National Scie... |

14 | Iterative methods for finding a trust-region step
- Erway, Gill, et al.
(Show Context)
Citation Context ...g the MSS method leads to fewer overall function evaluations for the overall trust-region method. This is consistent with other subproblem solvers that solve trust-region subproblems to high accuracy =-=[11, 10]-=-. Table III also suggests that while the MSS method took more time overall, it solved the subproblems within a comparable time frame. THE MSS METHOD 11 Table 2. MSS method and Steihaug-Toint method on... |

12 | Analysis on the conjugate gradient method
- Yuan
- 1993
(Show Context)
Citation Context .... In particular, the Steihaug-Toint method computes an approximate solution to (1) that is guaranteed to achieve at least half the optimal reduction in the quadratic function when the model is convex =-=[25, 15]-=-, but does not specifically seek to solve the minimization problem to high accuracy when the solution lies on the boundary. This paper presents an algorithm to solve (1) to any user-defined accuracy. ... |

8 | A subspace minimization method for the trustregion step
- Erway, Gill
(Show Context)
Citation Context ...g the MSS method leads to fewer overall function evaluations for the overall trust-region method. This is consistent with other subproblem solvers that solve trust-region subproblems to high accuracy =-=[11, 10]-=-. Table III also suggests that while the MSS method took more time overall, it solved the subproblems within a comparable time frame. THE MSS METHOD 11 Table 2. MSS method and Steihaug-Toint method on... |

6 | Solving the quadratic trustregion subproblem in a low-memory BFGS framework
- Apostolopoulou, Sotiropoulos, et al.
(Show Context)
Citation Context ...trix, and thus, avoid computing Choleksy factorizations. Like with [4], there are potential stability issues that are not addressed regarding inverting the M ×M matrix. Finally, Apostolopoulou et al. =-=[2, 1]-=- derive a closed-form expression for (B + σI)−1 to solve the first equation in (2). The authors are able to explicitly compute the eigenvalues of THE MSS METHOD 3 B, provided M = 1 [2, 1] or M = 2 [1]... |

5 | Notes on limited memory BFGS updating in a trust-region framework
- Burke, Weigmann
- 1997
(Show Context)
Citation Context ...mpute and store Cholesky factorizations for unstructured Hessians. Several researchers have proposed adaptations of the Moré-Sorensen direct method into the limited-memory BFGS setting. Burke et al. =-=[4]-=- derive a method via the Sherman-MorrisonWoodbury formula that uses two M ×M Cholesky factorizations, where M is the number of limited-memory updates. While this technique is able to exploit propertie... |

2 |
Limited-memory bfgs systems with diagonal updates. Linear Algebra and its
- Erway, Marcia
(Show Context)
Citation Context ...rensen direct method into an L-BFGS setting for large-scale optimization. The MSS method makes use of a recently proposed stable fast direct method for solving large shifted BFGS systems of equations =-=[13, 12]-=- and is able to compute solutions to any user-defined accuracy. This MATLAB implementation is a matrix-free iterative method for large-scale optimization. Numerical experiments on the CUTEr [3, 16]) s... |

2 |
A modified nearly exact method for solving low-rank trust region subproblem
- Lu, Monteiro
- 2004
(Show Context)
Citation Context ...technique is able to exploit properties of L-BFGS updates, there are potential instability issues related to their proposed use of the ShermanMorrison-Woodbury that are not addressed. Lu and Monteiro =-=[18]-=- also explore a MoréSorensen method implementation when B has special structure; namely, B = D + V EV T , where D and E are positive diagonal matrices, and V has a small number of columns. Their appr... |

1 | A practical method for solving large-scale TRS
- Apostolopoulou, Sotiropoulos, et al.
(Show Context)
Citation Context ...trix, and thus, avoid computing Choleksy factorizations. Like with [4], there are potential stability issues that are not addressed regarding inverting the M ×M matrix. Finally, Apostolopoulou et al. =-=[2, 1]-=- derive a closed-form expression for (B + σI)−1 to solve the first equation in (2). The authors are able to explicitly compute the eigenvalues of THE MSS METHOD 3 B, provided M = 1 [2, 1] or M = 2 [1]... |

1 | Shifted L-BFGS systems
- Erway, Jain, et al.
- 2012
(Show Context)
Citation Context ...rensen direct method into an L-BFGS setting for large-scale optimization. The MSS method makes use of a recently proposed stable fast direct method for solving large shifted BFGS systems of equations =-=[13, 12]-=- and is able to compute solutions to any user-defined accuracy. This MATLAB implementation is a matrix-free iterative method for large-scale optimization. Numerical experiments on the CUTEr [3, 16]) s... |