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Amenable cones: error bounds without constraint qualifications

  • Bruno F. LourençoEmail author
Full Length Paper Series A
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Abstract

We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The number of compositions is related to the facial reduction technique and the singularity degree of the problem. In particular, we show that symmetric cones are amenable and compute facial residual functions. From that, we are able to furnish a new Hölderian error bound, thus extending and shedding new light on an earlier result by Sturm on semidefinite matrices. We also provide error bounds for the intersection of amenable cones, this will be used to prove error bounds for the doubly nonnegative cone. At the end, we list some open problems.

Keywords

Error bounds Amenable cones Facial reduction Singularity degree Symmetric cones Feasibility problem Subtransversality 

Mathematics Subject Classification

90C31 65G99 17C55 

Notes

Acknowledgements

We thank the editors and four referees for their insightful comments, which helped to improve the paper substantially. In particular, the discussion on tangentially exposed cones and subtransversality was suggested by Referee 1. Also, comments from Referees 1 and 2 motivated Remark 10. We would like to thank Prof. Gábor Pataki for helpful advice and for suggesting that we take a look at projectionally exposed cones. Incidentally, this was also suggested by Referee 4. Referee 4 also suggested the remark on the tightness of the error bound for doubly nonnegative matrices. Feedback and encouragement from Prof. Tomonari Kitahara, Prof. Masakazu Muramatsu and Prof. Takashi Tsuchiya were highly helpful and they provided the official translation of “amenable cone” to Japanese:
(kyoujunsui). This work was partially supported by the Grant-in-Aid for Scientific Research (B) (18H03206) and the Grant-in-Aid for Young Scientists (19K20217) from Japan Society for the Promotion of Science.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan

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