### Citations

984 |
The Symmetric Eigenvalue Problem
- Parlett
- 1980
(Show Context)
Citation Context ...+1), which is always less than −λ1 and thus lies in the N region. This process is then repeated. Numerous standard results relating to both inverse iteration and the Rayleigh quotient may be found in =-=[9,36,43]-=-; for simplicity, these results typically assume that the eigenvalue for which convergence occurs is simple and that M = I. Lemmas 3.6 and 3.7 extend two of these results to the generalized eigenvalue... |

773 | Applied Numerical Linear Algebra
- Demmel
- 1997
(Show Context)
Citation Context ...+1), which is always less than −λ1 and thus lies in the N region. This process is then repeated. Numerous standard results relating to both inverse iteration and the Rayleigh quotient may be found in =-=[9,36,43]-=-; for simplicity, these results typically assume that the eigenvalue for which convergence occurs is simple and that M = I. Lemmas 3.6 and 3.7 extend two of these results to the generalized eigenvalue... |

445 |
Numerical Linear Algebra.
- Trefethen, Bau
- 1997
(Show Context)
Citation Context ...+1), which is always less than −λ1 and thus lies in the N region. This process is then repeated. Numerous standard results relating to both inverse iteration and the Rayleigh quotient may be found in =-=[9,36,43]-=-; for simplicity, these results typically assume that the eigenvalue for which convergence occurs is simple and that M = I. Lemmas 3.6 and 3.7 extend two of these results to the generalized eigenvalue... |

386 | Benchmarking optimization software with performance profiles,
- Dolan, More
- 2002
(Show Context)
Citation Context ...problems and a set of competing algorithms, the i-th performance profiles pi(α) indicates the fraction of problems for which the i-th algorithm is within a factor α of the best for a given metric—see =-=[10]-=- for a formal definition of performance profiles and a discussion of their properties.s0s0.2s0.4s0.6s0.8s1s1s1.5s2s2.5s3s3.5 dgqt TRS α p(α) Figure 4.1: Performance profile for the numbers of factoriz... |

308 |
Computing a trust region step
- Moré, Sorensen
- 1983
(Show Context)
Citation Context ...r given λ is both feasible and efficient. Our intention here is to revisit the possibility of solving our problems using factorization, and in particular to reassess the pioneering Gay-Moré-Sorensen =-=[17, 34]-=- methods in the light of modern sparse factorization. In §2 we discuss optimality conditions for our two subproblems and see how they lead to a robust framework for their solution. Details are given i... |

306 |
The multifrontal solution of indefinite sparse symmetric linear equations,
- Duff
- 1983
(Show Context)
Citation Context ...equation solvers—such methods are generally reliable and effective [19, 23]. We use the commercial package1 MA57 [12] but provide a slightly-less effective alternative SILS (based on the earlier MA27 =-=[13]-=-) for those unable to access MA57. 3.2 The secular function and its properties Suppose that x(λ) satisfies (2.3). We consider properties of the secular function π(λ) def = xT (λ)Mx(λ) ≡ ‖x(λ)‖2M . (3.... |

295 |
Optimization Algorithms on Matrix Manifolds.
- Absil, Mahony, et al.
- 2008
(Show Context)
Citation Context ...ion to replace the trust region constraint in TRS by the equation ‖x‖M = ∆ since there is currently much interest in solving optimization problems on Riemannian manifolds including the hyperellipsoid =-=[1, 2]-=-; in this case there is no longer the requirement that λ be positive, merely that λ ≥ −λ1, and the algorithm is adapted in the obvious way. Currently the possible improvement when λC ∈ N mentioned in ... |

259 | Mathematical Analysis, - Apostol - 1974 |

216 | Iterative Methods for the Solution of Equations - Traub - 1997 |

185 |
The conjugate gradient method and trust regions in large scale optimization,
- Steihaug
- 1983
(Show Context)
Citation Context ... mentioned for applications involving Hölder- but not Lipschitz-continuous derivatives [24]. Although it is now common to try to find approximate solutions to (1.1) and (1.2) using iterative methods =-=[5, 14, 15, 20, 25, 40, 41]-=-, there are still many problems for which a factorization of H + λM for given λ is both feasible and efficient. Our intention here is to revisit the possibility of solving our problems using factoriza... |

102 |
Recent developments in algorithms and software for trust region methods.
- More
- 1982
(Show Context)
Citation Context ...Hx+ σ p ‖x‖pM , (1.2) where the M-norm of x is ‖x‖M def= √ xTMx. Both problems arise as subproblems in unconstrained optimization; problem (1.1) occurs when computing the step in trust-region methods =-=[8, 33]-=-, while (1.2) plays the same role in more recent regularisation approaches [5,24,35,44]; for the latter p = 3 is by far the most common choice, although p < 3 has been mentioned for applications invol... |

99 |
Computing optimal locally constrained step,
- Gay
- 1981
(Show Context)
Citation Context ...r given λ is both feasible and efficient. Our intention here is to revisit the possibility of solving our problems using factorization, and in particular to reassess the pioneering Gay-Moré-Sorensen =-=[17, 34]-=- methods in the light of modern sparse factorization. In §2 we discuss optimality conditions for our two subproblems and see how they lead to a robust framework for their solution. Details are given i... |

91 |
Trust-region methods.
- Toint
- 2000
(Show Context)
Citation Context ...Hx+ σ p ‖x‖pM , (1.2) where the M-norm of x is ‖x‖M def= √ xTMx. Both problems arise as subproblems in unconstrained optimization; problem (1.1) occurs when computing the step in trust-region methods =-=[8, 33]-=-, while (1.2) plays the same role in more recent regularisation approaches [5,24,35,44]; for the latter p = 3 is by far the most common choice, although p < 3 has been mentioned for applications invol... |

69 |
Towards an efficient sparsity exploiting Newton method for minimization,
- Toint
- 1981
(Show Context)
Citation Context ... mentioned for applications involving Hölder- but not Lipschitz-continuous derivatives [24]. Although it is now common to try to find approximate solutions to (1.1) and (1.2) using iterative methods =-=[5, 14, 15, 20, 25, 40, 41]-=-, there are still many problems for which a factorization of H + λM for given λ is both feasible and efficient. Our intention here is to revisit the possibility of solving our problems using factoriza... |

68 | Cubic regularization of Newton method and its global performance. - Nesterov, Polyak - 2006 |

65 |
CUTEr and SifDec: A constrained and unconstrained testing environment, revisited,
- Toint
- 2003
(Show Context)
Citation Context ...ate well the new design features we have added. To see more generally the effect of improved convergence in both easy and hard cases, we consider all the unconstrained problems contained in the CUTEr =-=[21]-=- test set; we restrict our attention to those problems involving 2000 or fewer variables, since the dense Cholesky factorization used by dgqt struggles with larger cases, and this leads to 97 examples... |

62 | Trust-region methods on Riemannian manifolds.
- Absil, Baker, et al.
- 2007
(Show Context)
Citation Context ...ion to replace the trust region constraint in TRS by the equation ‖x‖M = ∆ since there is currently much interest in solving optimization problems on Riemannian manifolds including the hyperellipsoid =-=[1, 2]-=-; in this case there is no longer the requirement that λ be positive, merely that λ ≥ −λ1, and the algorithm is adapted in the obvious way. Currently the possible improvement when λC ∈ N mentioned in ... |

55 |
On Practical Conditions for the Existence and Uniqueness of Solutions to the General Equality Quadratic Programming problem
- Gould
- 1985
(Show Context)
Citation Context ...note that λ ∈ F if and only if the matrix( H + λM AT A 0 ) (5.2) is positive definite in the null-space of A, or equivalently that (5.2) is non-singular and has precisely rank(A) negative eigenvalues =-=[7, 18]-=-. To verify the latter condition and then to solve (5.1), any inertia-revealing symmetric, indefinite factorization package is appropriate (see §3.1), although now numerical pivoting will be required ... |

47 |
MA57—a code for the solution of sparse symmetric definite and indefinite systems.
- DUFF
- 2004
(Show Context)
Citation Context ... occurs). These features are common to a number of well-known sparse, symmetric linear equation solvers—such methods are generally reliable and effective [19, 23]. We use the commercial package1 MA57 =-=[12]-=- but provide a slightly-less effective alternative SILS (based on the earlier MA27 [13]) for those unable to access MA57. 3.2 The secular function and its properties Suppose that x(λ) satisfies (2.3).... |

47 | A numerical evaluation of sparse direct solvers for the solution of large sparse, symmetric linear systems of equations
- Gould, Hu, et al.
- 2005
(Show Context)
Citation Context ...and immediately stop the factorization if this occurs). These features are common to a number of well-known sparse, symmetric linear equation solvers—such methods are generally reliable and effective =-=[19, 23]-=-. We use the commercial package1 MA57 [12] but provide a slightly-less effective alternative SILS (based on the earlier MA27 [13]) for those unable to access MA57. 3.2 The secular function and its pro... |

44 | Minimizing a quadratic over a sphere,
- Hager
- 2001
(Show Context)
Citation Context ... mentioned for applications involving Hölder- but not Lipschitz-continuous derivatives [24]. Although it is now common to try to find approximate solutions to (1.1) and (1.2) using iterative methods =-=[5, 14, 15, 20, 25, 40, 41]-=-, there are still many problems for which a factorization of H + λM for given λ is both feasible and efficient. Our intention here is to revisit the possibility of solving our problems using factoriza... |

42 |
Solving the trust-region subproblem using the Lanczos method,
- Toint
- 1999
(Show Context)
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38 |
The modification of Newton’s method for unconstrained optimization by bounding cubic terms.
- Griewank
- 1981
(Show Context)
Citation Context ...se as subproblems in unconstrained optimization; problem (1.1) occurs when computing the step in trust-region methods [8, 33], while (1.2) plays the same role in more recent regularisation approaches =-=[5,24,35,44]-=-; for the latter p = 3 is by far the most common choice, although p < 3 has been mentioned for applications involving Hölder- but not Lipschitz-continuous derivatives [24]. Although it is now common ... |

36 |
GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization,
- Toint
- 2003
(Show Context)
Citation Context ...oped in this paper have been implemented as a pair of thread-safe Fortran 95 packages—respectively TRS and RQS for problems (1.1) and (1.2)—as part of version 2.3 of the GALAHAD optimization library2 =-=[22]-=-. The packages provide a number of options. 2Available from http://galahad.rl.ac.uk/galahad-www/. On solving trust-region and other regularised subproblems in optimization 29 The matrix H (and optiona... |

36 | An algorithm for minimization using exact second derivatives.
- Hebden
- 1973
(Show Context)
Citation Context ...+ x2 within a ℓ2-norm trust region of radius 4 (“easy” case, left) and those for − 1 4 x21 − 18x22 + x2 within a trust region of radius 5 (“hard” case, right). the poles present at λ = −λi when β > 0 =-=[26,38]-=-. With fast ultimate convergence assured in the easy case, the art is thus to be able to find an initial λ ∈ L. We return to this in §3. The hard case may happen when uTi c ≡ uTi c = 0 for all i for w... |

33 |
Adaptive cubic overestimation methods for unconstrained optimization. Part II: worst-case function-evaluation complexity.
- Toint
- 2011
(Show Context)
Citation Context ...se as subproblems in unconstrained optimization; problem (1.1) occurs when computing the step in trust-region methods [8, 33], while (1.2) plays the same role in more recent regularisation approaches =-=[5,24,35,44]-=-; for the latter p = 3 is by far the most common choice, although p < 3 has been mentioned for applications involving Hölder- but not Lipschitz-continuous derivatives [24]. Although it is now common ... |

31 | A numerical evaluation of HSL packages for the direct solution of large sparse, symmetric linear systems of equations
- Gould, Scott
(Show Context)
Citation Context ...and immediately stop the factorization if this occurs). These features are common to a number of well-known sparse, symmetric linear equation solvers—such methods are generally reliable and effective =-=[19, 23]-=-. We use the commercial package1 MA57 [12] but provide a slightly-less effective alternative SILS (based on the earlier MA27 [13]) for those unable to access MA57. 3.2 The secular function and its pro... |

30 | Affine conjugate adaptive Newton methods for nonlinear elastomechanics.
- Weiser, Deuflhard, et al.
- 2007
(Show Context)
Citation Context ...se as subproblems in unconstrained optimization; problem (1.1) occurs when computing the step in trust-region methods [8, 33], while (1.2) plays the same role in more recent regularisation approaches =-=[5,24,35,44]-=-; for the latter p = 3 is by far the most common choice, although p < 3 has been mentioned for applications involving Hölder- but not Lipschitz-continuous derivatives [24]. Although it is now common ... |

23 | An out-of-core sparse Cholesky solver
- Reid, Scott
(Show Context)
Citation Context ...actorization technology has advanced rapidly of late, and both parallel/multi-core and out-of-core factorizations are now available and capable of coping with matrices of high (in the millions) order =-=[3, 27, 28, 37, 39]-=-. Some iterative methods [5, 20] for the solution of (1.1) and (1.2) solve sequences of problems of the same form, albeit now with simpler tridiagonal matrices H . Clearly the improvements suggested i... |

16 | Numerical solution of a secular equation. - Melman - 1995 |

16 | A unifying convergence analysis of second-order methods for secular equations. - Melman - 1997 |

15 |
Smoothing by spline functions II,
- Reinsch
- 1971
(Show Context)
Citation Context ...+ x2 within a ℓ2-norm trust region of radius 4 (“easy” case, left) and those for − 1 4 x21 − 18x22 + x2 within a trust region of radius 5 (“hard” case, right). the poles present at λ = −λi when β > 0 =-=[26,38]-=-. With fast ultimate convergence assured in the easy case, the art is thus to be able to find an initial λ ∈ L. We return to this in §3. The hard case may happen when uTi c ≡ uTi c = 0 for all i for w... |

14 | Adapting a parallel sparse direct solver to architectures with clusters of SMPs
- Amestoy, Duff, et al.
- 2003
(Show Context)
Citation Context ...actorization technology has advanced rapidly of late, and both parallel/multi-core and out-of-core factorizations are now available and capable of coping with matrices of high (in the millions) order =-=[3, 27, 28, 37, 39]-=-. Some iterative methods [5, 20] for the solution of (1.1) and (1.2) solve sequences of problems of the same form, albeit now with simpler tridiagonal matrices H . Clearly the improvements suggested i... |

14 | Iterative methods for finding a trust-region step
- Erway, Gill, et al.
(Show Context)
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13 | Trust-region and other regularisation of linear least-squares problems - Toint |

12 |
Definiteness and semidefiniteness of quadratic forms revisited
- Chabrillac, Crouzeix
- 1984
(Show Context)
Citation Context ...note that λ ∈ F if and only if the matrix( H + λM AT A 0 ) (5.2) is positive definite in the null-space of A, or equivalently that (5.2) is non-singular and has precisely rank(A) negative eigenvalues =-=[7, 18]-=-. To verify the latter condition and then to solve (5.1), any inertia-revealing symmetric, indefinite factorization package is appropriate (see §3.1), although now numerical pivoting will be required ... |

10 | On the linear least squares problem with a quadratic constraint, - Gander - 1978 |

10 | Algorithmic performance studies on graphics processing units." Journal of parallel and distributed computing 68(10
- Schenk, Christen
- 2008
(Show Context)
Citation Context ...actorization technology has advanced rapidly of late, and both parallel/multi-core and out-of-core factorizations are now available and capable of coping with matrices of high (in the millions) order =-=[3, 27, 28, 37, 39]-=-. Some iterative methods [5, 20] for the solution of (1.1) and (1.2) solve sequences of problems of the same form, albeit now with simpler tridiagonal matrices H . Clearly the improvements suggested i... |

8 | A subspace minimization method for the trustregion step
- Erway, Gill
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7 |
A DAG-based Parallel Cholesky Factorization for Multicore Systems
- Hogg
- 2008
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6 | A numerical Comparison of Methods for Solving Secular Equations - Melman - 1997 |

4 | Analysis of third-order methods for secular equations - Melman - 1998 |

3 | On Taylor series approximations for trust-region and regularized subproblems in optimization - Dollar - 2009 |

1 |
Sparse symmetric linear solvers for multicore architectures
- Hogg, Reid, et al.
- 2009
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