## On the Convergence of Successive Linear Programming Algorithms (2003)

Citations: | 4 - 1 self |

### BibTeX

@TECHREPORT{Byrd03onthe,

author = {Richard H. Byrd and Nicholas I. M. Gould and Jorge Nocedal and Richard A. Waltz},

title = {On the Convergence of Successive Linear Programming Algorithms},

institution = {},

year = {2003}

}

### OpenURL

### Abstract

We analyze the global convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more speci cally the SLP-EQP approach presented in [1]. Every iteration requires the solution of two trust region subproblems involving linear and quadratic models, respectively. The interaction between the trust regions of these subproblems requires careful consideration. It is shown under mild assumptions that there exist an accumulation point which is a critical point for the penalty function.

### Citations

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(Show Context)
Citation Context ...much reduction in the quadratic model as at its Cauchy point, but also allows the step to expand into a possibly enlarged master trust region. Steps 3 and 4 are standard trust-region acceptance rules =-=[-=-3]. The ratio k of the actual to the predicted reduction of is used as a step acceptance criterion. If this ratio is negative, or close to zero, the step is rejected and the overall trust-region rad... |

32 |
de la Maza. Nonlinear programming and nonsmooth optimization by successive linear programming
- Fletcher, Sainz
- 1989
(Show Context)
Citation Context ... a set of equality constraints given by the working set, and subject to a trust region bound. A particular instance of this approach is the SLP-EQP algorithm proposed by Fletcher and Sainz de la Maza =-=[7]-=-. The main purpose of this article is to establish the global convergence of this class of penalty methods. The analysis will be phrased in the general context of composite nonsmooth optimization prob... |

26 | An exact potential method for constrained maxima - Pietrzykowski - 1968 |

19 |
On the superlinear convergence of a trust region algorithm for nonsmooth optimization
- Yuan
- 1985
(Show Context)
Citation Context ...s much promise. In the next section we describe the algorithm to be analyzed, and in x3 we present the global convergence results. We note that the theory of non-smooth optimization developed by Yuan =-=[12, 1-=-4] cannot be applied because in our algorithms the two trust regions in uence each other, whereas Yuan assumes that a single trust region is used. The analysis presented here is signicantly dierent fr... |

16 |
On the convergence of a new trust region algorithm
- Yuan
- 1995
(Show Context)
Citation Context ...s much promise. In the next section we describe the algorithm to be analyzed, and in x3 we present the global convergence results. We note that the theory of non-smooth optimization developed by Yuan =-=[12, 1-=-4] cannot be applied because in our algorithms the two trust regions in uence each other, whereas Yuan assumes that a single trust region is used. The analysis presented here is signicantly dierent fr... |

13 |
Practical Methods of Optimization: Constrained Optimization, volume 2
- Fletcher
- 1981
(Show Context)
Citation Context ...ithms 3 where F i (x) = f i (x), i = 1; : : : ; p are smooth functions of x, and ! : IR p ! IR is convex but may be nonsmooth. Such problems have been considered by a number of authors over the years =-=[5, 6, 7, 9, 10, 11, 12, 13]-=-. The penalty function (1:2) used to solve the nonlinear program is a special case of (2:1) obtained when E \I = ; and E [I = f2; : : : ; pg by setting f 1 (x) = f(x); f i (x) = h i (x); i 2 E ; f i (... |

10 |
A penalty function method converging directly to a constrained optimum
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- 1977
(Show Context)
Citation Context ...e continuously dierentiable. Our interest is in the case where there are a large number of unknowns. The class of algorithms studied in this paper solve (1:1) by minimizing an exact penalty function [4, 8] of the form (x; ) = f(x) + kh(x)k + kg (x)k; (1.2) where k k is a (monotonic) norm, g i (x) = min(g i (x); 0); and > 0 is a parameter which is adaptively chosen so that critical points of (1:... |

9 |
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(Show Context)
Citation Context ...ithms 3 where F i (x) = f i (x), i = 1; : : : ; p are smooth functions of x, and ! : IR p ! IR is convex but may be nonsmooth. Such problems have been considered by a number of authors over the years =-=[5, 6, 7, 9, 10, 11, 12, 13]-=-. The penalty function (1:2) used to solve the nonlinear program is a special case of (2:1) obtained when E \I = ; and E [I = f2; : : : ; pg by setting f 1 (x) = f(x); f i (x) = h i (x); i 2 E ; f i (... |

4 |
An active set algorithm for nonlinear programming using linear programming and equality constrained subproblems
- Byrd, Gould, et al.
- 2002
(Show Context)
Citation Context ... convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more specically the SLP-EQP approach presented in [=-=1]-=-. Every iteration requires the solution of two trust region subproblems involving linear and quadratic models, respectively. The interaction between the trust regions of these subproblems requires car... |

4 | On the global convergence of an SLP- algorithm that takes EQP steps. Numerical Analysis Report NA/199 - Chin, Fletcher - 1999 |

3 |
A model algorithm for composite nondierentiable optimization problems. Mathematical Programming Studies
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(Show Context)
Citation Context ...ithms 3 where F i (x) = f i (x), i = 1; : : : ; p are smooth functions of x, and ! : IR p ! IR is convex but may be nonsmooth. Such problems have been considered by a number of authors over the years =-=[5, 6, 7, 9, 10, 11, 12, 13]-=-. The penalty function (1:2) used to solve the nonlinear program is a special case of (2:1) obtained when E \I = ; and E [I = f2; : : : ; pg by setting f 1 (x) = f(x); f i (x) = h i (x); i 2 E ; f i (... |

3 |
An example of only linear convergence of trust region algorithms for nonsmooth optimization
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Citation Context |

1 |
An inexact algorithm for composite nondierentiable optimization
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