Strong Duality in Conic Linear Programming: Facial Reduction and Extended Duals

  • Gábor Pataki
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 50)


The facial reduction algorithm (FRA) of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program
$$\displaystyle{ \sup \,\{\,\langle c,x\rangle \,\vert \,Ax \leq _{K}b\,\} }$$
in the absence of any constraint qualification. The FRA solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana’s dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself. Ramana, Tunçel, and Wolkowicz showed that these approaches are closely related; in particular, they proved the correctness of Ramana’s dual using certificates from a facial reduction algorithm. Here we give a simple and self-contained exposition of facial reduction, of extended duals, and generalize Ramana’s dual:
  • We state a simple FRA and prove its correctness.

  • Building on this algorithm we construct a family of extended duals when K is a nice cone. This class of cones includes the semidefinite cone and other important cones.

Key words

Conic linear programming Strong duality Semidefinite programming Facial reduction Extended duals Nice cones 

Mathematics Subject Classifications (2010)

Primary: 90C46 49N15 90C22 90C25 secondary: 52A40 52A41 



I would like to thank an anonymous referee and Minghui Liu for their helpful comments.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of North Carolina at Chapel HillChapel HillUSA

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