License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.78
URN: urn:nbn:de:0030-drops-63368
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6336/
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Kurpisz, Adam ; Leppänen, Samuli ; Mastrolilli, Monaldo

Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems

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LIPIcs-ICALP-2016-78.pdf (0.5 MB)

Abstract

We give two results concerning the power of the Sum-Of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that (n+2d-1)/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.

Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is n-1. We disprove this conjecture and derive lower and upper bounds for the rank.

BibTeX - Entry

@InProceedings{kurpisz_et_al:LIPIcs:2016:6336,
  author =	{Adam Kurpisz and Samuli Lepp{\"a}nen and Monaldo Mastrolilli},
  title =	{{Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{78:1--78:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6336},
  URN =		{urn:nbn:de:0030-drops-63368},
  doi =		{10.4230/LIPIcs.ICALP.2016.78},
  annote =	{Keywords: SoS/Lasserre hierarchy, lift and project methods, binary polynomial optimization}
}

Keywords: SoS/Lasserre hierarchy, lift and project methods, binary polynomial optimization
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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