An interior-point ℓ1-penalty method for nonlinear optimization

Nick I M Gould, Dominique Orban, Philippe L. Toint

Research output: Contribution in Book/Catalog/Report/Conference proceedingChapter (peer-reviewed)

Abstract

We describe a mixed interior/exterior-point method for nonlinear programming that handles constraints by way of an ℓ1-penalty function. The penalty problem is reformulated as a smooth inequality-constrained problem that always possesses bounded multipliers, and that may be solved using interior-point techniques as finding a strictly feasible point is trivial. If finite multipliers exist for the original problem, exactness of the penalty function eliminates the need to drive the penalty parameter to infinity. If the penalty parameter needs to increase without bound and if feasibility is ultimately attained, a certificate of degeneracy is delivered. Global and fast local convergence of the proposed scheme are established and practical aspects of the method are discussed.

Original languageEnglish
Title of host publicationSpringer Proceedings in Mathematics and Statistics
Subtitle of host publicationProceedings of NAOIII 2014
PublisherSpringer New York
Pages117-150
Number of pages34
Volume134
ISBN (Print)9783319176888
DOIs
Publication statusPublished - 2015
Event3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014 - Muscat, Oman
Duration: 5 Jan 20149 Jan 2014

Conference

Conference3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014
CountryOman
CityMuscat
Period5/01/149/01/14

Fingerprint

Penalty Method
Interior Point
Nonlinear Optimization
Penalty
Penalty Function
Multiplier
Exactness
Local Convergence
Certificate
Degeneracy
Nonlinear Programming
Trivial
Interior
Eliminate
Strictly
Infinity

Keywords

  • Elastic variables
  • Interior point
  • Nonconvex Optimization
  • ℓ-Penalty

Cite this

Gould, N. I. M., Orban, D., & Toint, P. L. (2015). An interior-point ℓ1-penalty method for nonlinear optimization. In Springer Proceedings in Mathematics and Statistics: Proceedings of NAOIII 2014 (Vol. 134, pp. 117-150). Springer New York. https://doi.org/10.1007/978-3-319-17689-5_6
Gould, Nick I M ; Orban, Dominique ; Toint, Philippe L. / An interior-point ℓ1-penalty method for nonlinear optimization. Springer Proceedings in Mathematics and Statistics: Proceedings of NAOIII 2014. Vol. 134 Springer New York, 2015. pp. 117-150
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Gould, NIM, Orban, D & Toint, PL 2015, An interior-point ℓ1-penalty method for nonlinear optimization. in Springer Proceedings in Mathematics and Statistics: Proceedings of NAOIII 2014. vol. 134, Springer New York, pp. 117-150, 3rd International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOIII-2014, Muscat, Oman, 5/01/14. https://doi.org/10.1007/978-3-319-17689-5_6

An interior-point ℓ1-penalty method for nonlinear optimization. / Gould, Nick I M; Orban, Dominique; Toint, Philippe L.

Springer Proceedings in Mathematics and Statistics: Proceedings of NAOIII 2014. Vol. 134 Springer New York, 2015. p. 117-150.

Research output: Contribution in Book/Catalog/Report/Conference proceedingChapter (peer-reviewed)

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AB - We describe a mixed interior/exterior-point method for nonlinear programming that handles constraints by way of an ℓ1-penalty function. The penalty problem is reformulated as a smooth inequality-constrained problem that always possesses bounded multipliers, and that may be solved using interior-point techniques as finding a strictly feasible point is trivial. If finite multipliers exist for the original problem, exactness of the penalty function eliminates the need to drive the penalty parameter to infinity. If the penalty parameter needs to increase without bound and if feasibility is ultimately attained, a certificate of degeneracy is delivered. Global and fast local convergence of the proposed scheme are established and practical aspects of the method are discussed.

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Gould NIM, Orban D, Toint PL. An interior-point ℓ1-penalty method for nonlinear optimization. In Springer Proceedings in Mathematics and Statistics: Proceedings of NAOIII 2014. Vol. 134. Springer New York. 2015. p. 117-150 https://doi.org/10.1007/978-3-319-17689-5_6