Mathematical Programming Computation

, Volume 11, Issue 3, pp 503–586 | Cite as

Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

  • Yuzixuan Zhu
  • Gábor PatakiEmail author
  • Quoc Tran-Dinh
Full Length Paper


We introduce Sieve-SDP, a simple facial reduction algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP inspects the constraints of the problem to detect lack of strict feasibility, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on several problem collections from the literature, with many SDPs coming from polynomial optimization.


Semidefinite programming Preprocessing Strict feasibility Strong duality Facial reduction Polynomial optimization 

Mathematics Subject Classification

90-08 90C22 90C25 90C06 



We thank the Technical Editor and the referees for their helpful comments. The second author, Gábor Pataki, is supported by the National Science Foundation, Award DMS-1817272. The third author, Quoc Tran-Dinh, is supported in part by the National Science Foundation, Award DMS-1619884. We are very grateful to Erling Andersen at Mosek for running several SDPs, and explaining the results; to Joachim Dahl at Mosek for helpful discussions on converting SDPs, and for providing his conversion code; to Didier Henrion and Kim-Chuan Toh for providing us with some of the datasets; to Frank Permenter and Johan Löfberg for helpful comments; to Oktay Günlük for helping us to invent the name “Sieve-SDP”; and to Hans Mittelmann for helping us with some of the large-scale SDPs.

Supplementary material


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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 Statistics and Operations ResearchThe University of North Carolina at Chapel HillChapel HillUSA

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