## Interior methods for nonlinear optimization (2002)

Venue: | SIAM Review |

Citations: | 76 - 4 self |

### BibTeX

@ARTICLE{Forsgren02interiormethods,

author = {Anders Forsgren and Philip E. Gill and Margaret H. Wright},

title = {Interior methods for nonlinear optimization},

journal = {SIAM Review},

year = {2002},

volume = {44},

pages = {525--597}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

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Citation Context ...ation (6.6) were applied directly to (6.9). Interior-point algorithms based on the quadratic programming subproblem (6.17) typically utilize SQP-based merit functions to enforce convergence (see,e.g.,=-=[15,27,46,83,123]-=-). If indefiniteness is addressed with a suitable line-search or trust-region technique,the limit points of the sequence of iterates satisfy the second-order necessary optimality conditions for (6.9) ... |

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30 |
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28 |
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Citation Context ...HC (except when the constraints are simple bounds,in which case HC is simply H plus a positive diagonal matrix). An alternative strategy is to factorize the full (n+m)×(n+m) system in (5.2) (see,e.g.,=-=[43,49,50]-=-),typically after symmetrizing the system. A symmetric matrix can be created by multiplying the second block of equations in (5.2) by Λ−1 and changing the sign of the second block of columns,giving (5... |

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28 |
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Citation Context ...ion of a linear program (2.10); verifying assumption (iii) for all p such that J ∗ A p ≥ 0 requires finding the global minimizer of a possibly indefinite quadratic form over a cone,an NP-hard problem =-=[75,82]-=-,not to mention the issue of how to check that (iii) holds for all λ ∈Mλ. If,however,the gradients of the active constraints at x ∗ are linearly independent and strict complementarity holds,Theorem 2.... |

26 | Global convergence of a trust-region SQP-filter algorithms for nonlinear programming
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(Show Context)
Citation Context ...d to the filter,and all filter entries that it dominates are removed. In practice many details need to be specified in order to define a convergent filter algorithm; we refer the interested reader to =-=[35,36]-=-. Recently,several algorithms were proposed that use a filter to force convergence of a primal-dual interior method (see [8,100,107]). In the primal-dual context,there are several ways to define the f... |