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## An interior-point affine-scaling trust-region method for semismooth equations with box constraints

Venue: | Comput. Optim. Appl |

Citations: | 3 - 0 self |

### Citations

1665 |
Optimization and Nonsmooth Analysis,
- Clarke
- 1983
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Citation Context ... Furthermore, given a mapping G : Rn → Rm, we write G′(x) for the Jacobian of G at x (if G is differentiable), and ∂G(x) for the generalized Jacobian of G at x (if G is locally Lipschitz), see Clarke =-=[9]-=-. Note, in particular, that ∂G(x) = {G′(x)} if G is continuously differentiable in x. We often use the short-hand notation Gk for the mapping G evaluated at a point xk ∈ Rn. Finally, several propertie... |

1345 |
Numerical methods for unconstrained optimization and nonlinear equations
- Dennis, Schnabel
- 1983
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Citation Context ...r bounds satisfy the relation −∞ ≤ li < ui ≤ +∞ for all i = 1, . . . , n. The corresponding unconstrained problem with Ω = Rn and F continuously differentiable is discussed in several books including =-=[1, 16, 18, 31, 32, 40]-=-. Extensions of the classical Newton method for the solution of the unconstrained problem with F being semismooth may be found in [43, 42, 41, 22]. Despite the popularity of the unconstrained system o... |

959 | Introduction to Numerical Analysis - Stoer, Bulirsch - 1980 |

757 |
Iterative Solutions of Nonlinear Equations
- Ortega, Rheinboldt
- 1970
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Citation Context ...r bounds satisfy the relation −∞ ≤ li < ui ≤ +∞ for all i = 1, . . . , n. The corresponding unconstrained problem with Ω = Rn and F continuously differentiable is discussed in several books including =-=[1, 16, 18, 31, 32, 40]-=-. Extensions of the classical Newton method for the solution of the unconstrained problem with F being semismooth may be found in [43, 42, 41, 22]. Despite the popularity of the unconstrained system o... |

572 | A limited memory algorithm for bound constrained optimization.
- Byrd, Lu, et al.
- 1995
(Show Context)
Citation Context ... 3 While there are not so many papers dealing with problem (1), there is a huge literature with different methods for the solution of box constrained optimization problems, and we refer the reader to =-=[6, 7, 10, 11, 12, 13, 17, 20, 21, 26, 27, 30, 34, 35, 36, 45, 48, 49, 50]-=-, for example. These references include active set-type methods, projected Newton methods, penalty methods, pattern search, and affine-scaling interior-point methods. The latter class of methods is of... |

553 |
Iterative Methods for Linear and Nonlinear Equations,
- Kelley
- 1995
(Show Context)
Citation Context ...r bounds satisfy the relation −∞ ≤ li < ui ≤ +∞ for all i = 1, . . . , n. The corresponding unconstrained problem with Ω = Rn and F continuously differentiable is discussed in several books including =-=[1, 16, 18, 31, 32, 40]-=-. Extensions of the classical Newton method for the solution of the unconstrained problem with F being semismooth may be found in [43, 42, 41, 22]. Despite the popularity of the unconstrained system o... |

377 |
An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds,”
- Coleman, Li
- 1996
(Show Context)
Citation Context ... 3 While there are not so many papers dealing with problem (1), there is a huge literature with different methods for the solution of box constrained optimization problems, and we refer the reader to =-=[6, 7, 10, 11, 12, 13, 17, 20, 21, 26, 27, 30, 34, 35, 36, 45, 48, 49, 50]-=-, for example. These references include active set-type methods, projected Newton methods, penalty methods, pattern search, and affine-scaling interior-point methods. The latter class of methods is of... |

327 |
A nonsmooth version of Newton’s method,
- Qi, Sun
- 1993
(Show Context)
Citation Context ...tiable is discussed in several books including [1, 16, 18, 31, 32, 40]. Extensions of the classical Newton method for the solution of the unconstrained problem with F being semismooth may be found in =-=[43, 42, 41, 22]-=-. Despite the popularity of the unconstrained system of nonlinear equations, the number of references dealing with the box constrained problem (1) (and with F being either smooth or nonsmooth) is stil... |

260 |
Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II.
- Facchinei, Pang
- 2003
(Show Context)
Citation Context ...tiable is discussed in several books including [1, 16, 18, 31, 32, 40]. Extensions of the classical Newton method for the solution of the unconstrained problem with F being semismooth may be found in =-=[43, 42, 41, 22]-=-. Despite the popularity of the unconstrained system of nonlinear equations, the number of references dealing with the box constrained problem (1) (and with F being either smooth or nonsmooth) is stil... |

239 |
Introduction to Numerical Continuation Methods,
- Allgower, Georg
- 1990
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223 |
Convergence analysis of some algorithms for solving nonsmooth equations.
- Qi
- 1993
(Show Context)
Citation Context ...tiable is discussed in several books including [1, 16, 18, 31, 32, 40]. Extensions of the classical Newton method for the solution of the unconstrained problem with F being semismooth may be found in =-=[43, 42, 41, 22]-=-. Despite the popularity of the unconstrained system of nonlinear equations, the number of references dealing with the box constrained problem (1) (and with F being either smooth or nonsmooth) is stil... |

203 | Trust-region methods,
- Conn, Gould, et al.
- 2000
(Show Context)
Citation Context ...s to prove a global convergence result for Algorithm 2.1. To this end, we need two more assumptions. The first is a boundedness assumption which is rather standard in trust-region methods (see, e.g., =-=[14]-=-), and the second one is a condition regarding the choice of the diagonal scaling matrix D(x). (B) The sequence {Hk} generated by Algorithm 2.1 is bounded. (C) The scaling matrix D(x) satisfies (4) an... |

195 | Engineering and Economic Applications of Complementarity Problems,"
- Ferris, Pang
- 1997
(Show Context)
Citation Context ... mapping F is not defined on certain parts of the boundary of Ω due to some logarithmic terms, for example. Difficulties of this kind occur quite frequently in some economic equilibrium problems, cf. =-=[24]-=-. Therefore, although we assume for our theoretical analysis that F is defined on the open neighbourhood O of Ω, we feel that it is quite important from a practical point of view that all iterates sta... |

157 |
Solving nonlinear equations with Newton’s method,” in Fundamentals of Algorithms,
- Kelley
- 2003
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143 | Projected Newton methods for optimization problems with simple constraints,
- Bertsekas
- 1982
(Show Context)
Citation Context ... 3 While there are not so many papers dealing with problem (1), there is a huge literature with different methods for the solution of box constrained optimization problems, and we refer the reader to =-=[6, 7, 10, 11, 12, 13, 17, 20, 21, 26, 27, 30, 34, 35, 36, 45, 48, 49, 50]-=-, for example. These references include active set-type methods, projected Newton methods, penalty methods, pattern search, and affine-scaling interior-point methods. The latter class of methods is of... |

131 |
Nonsmooth equations: motivation and algorithms,
- Pang, Qi
- 1993
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107 | Newton’s method for large bound-constrained optimization problems,”
- Lin, More
- 1999
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106 | A semismooth equation approach to the solution of nonlinear complementarity problems,”
- Luca, Facchinei, et al.
- 1996
(Show Context)
Citation Context ...ds if F itself is continuously differentiable. However, there are also some interesting situations where f is continuously differentiable although F is not differentiable (but semismooth), see, e.g., =-=[23, 15, 8]-=- for some examples in the context of complementarity problems. We exploit the relation between the two problems (1) and (2) and follow an idea by Coleman and Li [10, 11] who observed that the first or... |

106 | Pattern search algorithms for bound constrained minimization”.
- Lewis, Torczon
- 1999
(Show Context)
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91 | A new merit function for nonlinear complementarity problems and a related algorithm,
- Facchinei, Soares
- 1997
(Show Context)
Citation Context ...ds if F itself is continuously differentiable. However, there are also some interesting situations where f is continuously differentiable although F is not differentiable (but semismooth), see, e.g., =-=[23, 15, 8]-=- for some examples in the context of complementarity problems. We exploit the relation between the two problems (1) and (2) and follow an idea by Coleman and Li [10, 11] who observed that the first or... |

90 | MCPLIB: A Collection of Nonlinear Mixed Complementarity Problems",
- Dirkse, Ferris
- 1994
(Show Context)
Citation Context ...ons are possible for the slightly more general class of mixed complementarity problems, and a large number of (often very difficult) test problems of this class is given in the MCPLIB collection, see =-=[19]-=-. Since the standard starting points for these examples are sometimes on the boundary of the feasible region, we use the strategy from [47] and project the standard starting vector x0 of the MCPLIB on... |

81 | L-BFGS-B: Algorithm 778: L-BFGSB, FORTRAN routines for large scale bound constrained optimization, - Zhu, Byrd, et al. - 1997 |

72 | Global convergence of a class of trust region algorithms for optimization with simple bounds, - Conn, Gould, et al. - 1988 |

55 | Trust-region interior-point algorithms for minimization problems with simple bounds
- Dennis, Vicente
- 1996
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54 |
A new trust region algorithm for bound constrained minimization,
- Friedlander, Martnez, et al.
- 1994
(Show Context)
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53 |
On the convergence of interior–reflective Newton methods for nonlinear minimization subject to bounds,
- Coleman, Li
- 1994
(Show Context)
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52 | A penalized Fischer-Burmeister NCP-function.
- Chen, Chen, et al.
- 2000
(Show Context)
Citation Context ...ds if F itself is continuously differentiable. However, there are also some interesting situations where f is continuously differentiable although F is not differentiable (but semismooth), see, e.g., =-=[23, 15, 8]-=- for some examples in the context of complementarity problems. We exploit the relation between the two problems (1) and (2) and follow an idea by Coleman and Li [10, 11] who observed that the first or... |

47 |
Testing a class of methods for solving minimization problems with simple bounds on the variables.
- Conn, Gould, et al.
- 1988
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47 | Newton Methods for Nonlinear Problems, - Deuflhard - 2004 |

26 |
Chemical Equilibrium Systems as Numerical Test Problmes.
- Meintjes, Morgan
- 1990
(Show Context)
Citation Context ... 11 equations and variables is described. A solution of this system is only physically meaningful if all components of the solution are real and positive. Following [4] we augment the system given in =-=[38]-=- to the size of n = 11000 and use li := 0, ui := ∞ and x0i := 1 for i = 1, . . . , n. 5. Boundary value problems. Discretizing a boundary value problem leads to a nonlinear system of equations. If thi... |

24 |
Cosnard, Numerical solution of nonlinear equations
- More, Y
- 1979
(Show Context)
Citation Context ... approximate the function and the first derivative. In order to get positive function values, we set li := 0 for all odd i, li := −∞ for all even i and ui = ∞ for i = 1, . . . , n. BVP2 is taken from =-=[39]-=-. The discretization given in [39] leads to a nonlinear system that has a unique solution in the box defined by li = −0.5 and ui = 0 for i = 1, . . . , n. We set n = 500, use the given bounds and star... |

22 | Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption,
- Heinkenschloss, Ulbrich, et al.
- 1999
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22 | Nonmonotone trust-region methods for boundconstrained semismooth equations with applications to nonlinear mixed complementarity problems,”
- Ulbrich
- 2001
(Show Context)
Citation Context ...linear equations, the number of references dealing with the box constrained problem (1) (and with F being either smooth or nonsmooth) is still very limited. Currently, we are only aware of the papers =-=[2, 3, 4, 5, 28, 29, 33, 44, 47]-=-. Most of these papers, however, appeared during the last few years, and we believe that the box constrained problem (1) is of increasing interest and quite important for several reasons. In fact, in ... |

22 |
Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise bounds
- Ulbrich, Ulbrich, et al.
- 1999
(Show Context)
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17 | Superlinear convergence of affine-scaling interior-point newton methods for infinite-dimensional nonlinear problems with pointwise bounds
- Ulbrich, Ulbrich
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16 |
Convergence of trust region algorithm for optimization with bounds when strict complementarity does not hold
- Lescrenier
- 1991
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15 |
An affine scaling trust-region approach to bound-constrained nonlinear systems,
- Bellavia, Macconi, et al.
- 2003
(Show Context)
Citation Context ...linear equations, the number of references dealing with the box constrained problem (1) (and with F being either smooth or nonsmooth) is still very limited. Currently, we are only aware of the papers =-=[2, 3, 4, 5, 28, 29, 33, 44, 47]-=-. Most of these papers, however, appeared during the last few years, and we believe that the box constrained problem (1) is of increasing interest and quite important for several reasons. In fact, in ... |

15 | Strictly feasible equation-based methods for mixed complementarity problems,
- Kanzow
- 2001
(Show Context)
Citation Context ...linear equations, the number of references dealing with the box constrained problem (1) (and with F being either smooth or nonsmooth) is still very limited. Currently, we are only aware of the papers =-=[2, 3, 4, 5, 28, 29, 33, 44, 47]-=-. Most of these papers, however, appeared during the last few years, and we believe that the box constrained problem (1) is of increasing interest and quite important for several reasons. In fact, in ... |

13 |
Inexact trust region method for large sparse systems of nonlinear equations,
- Lukšan
- 1994
(Show Context)
Citation Context ... := 1 for all i and consider the cases c = 0.99, c = 0.9999, and c = 1 with n = 1000. 2. The seven-diagonal problem. This is a nonlinear system of equations of variable dimension that can be found in =-=[37]-=-. The Jacobians have the structure of band matrices, which allows us to consider high dimensional cases. The system has several solutions, so we use the bounds li := 0 and ui := ∞, i = 1, . . . , n, t... |

12 |
An active set newton algorithm for large-scale nonlinear programs with box constraints
- Facchinei, Júdice
- 1998
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12 | Solving nonlinear systems of equations with simple bounds,
- Kozakevich, Martınez, et al.
- 1997
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12 |
Family of Projected Descent Methods for Optimization Problems with Simple Bounds,”
- Schwartz, Polak
- 1997
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9 |
Adjiman et al.,Handbook of Test Problems in Local and Global Optimization,
- Floudas, Pardalos, et al.
- 1999
(Show Context)
Citation Context ...hm from [3] are able to solve all test examples with the exception of one example from the Floudas et al. collection. However, this problem is regarded as very challenging, even the solution given in =-=[25]-=- seems to be wrong. The behaviour of the two methods on the other examples is more or less similar, and there is no clear winner on this set of test problems. We next present our numerical results for... |

9 | On affine-scaling interior-point Newton methods for nonlinear minimization with bound constraints
- Kanzow, Klug
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8 |
STRSCNE: A scaled trust-region solver for constrained nonlinear equations,
- Bellavia, Macconi, et al.
- 2004
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7 | A truncated Newton algorithm for large scale box constrained optimization
- Facchinei, Lucidi, et al.
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6 | An active set-type Newton method for constrained nonlinear systems,
- Kanzow
- 2001
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5 |
An interior global method for nonlinear systems with simple bounds
- Bellavia, Morini
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5 |
Active-set projected trust-region algorithm for box-constrained nonsmooth equations,”
- Qi, Tong, et al.
- 2004
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2 |
A two-dimensional trust-region method for large scale bound-constrained nonlinear systems
- Bellavia, Macconi, et al.
(Show Context)
Citation Context ...thod can be applied to both continuously differentiable and semismooth systems of equations. Hence the method is applicable to a wider class of problems than, for example, the related algorithms from =-=[2, 3, 4, 5]-=-, and this was illustrated for the class of complementarity problems. Compared to the method from the previous references, we also avoid a nonsingularity assumption that was used in these references i... |