Large-scale semidefinite programming via a saddle point Mirror-Prox algorithm

Zhaosong Lu, Arkadi Nemirovski, Renato D.C. Monteiro

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


In this paper, we first demonstrate that positive semidefiniteness of a large well-structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked, in a linear fashion, to the matrix. We derive also the "dual counterpart" of the outlined representation, which expresses the possibility of positive semidefinite completion of a well-structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix. Using the representations, we then reformulate well-structured large-scale semidefinite problems into smooth convex-concave saddle point problems, which can be solved by a Prox-method developed in [6] with efficiency script O sign (ε-1). Implementations and some numerical results for large-scale Lovász capacity and MAXCUT problems are finally presented.

Original languageEnglish (US)
Pages (from-to)211-237
Number of pages27
JournalMathematical Programming
Issue number2-3
StatePublished - Mar 1 2007
Externally publishedYes


  • Mirror-Prox method
  • Saddle point problem
  • Semidefinite programming


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