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

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Computational and Analytical Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

Abstract

The facial reduction algorithm (FRA) of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program

$$\displaystyle{ \sup \,\{\,\langle c,x\rangle \,\vert \,Ax \leq _{K}b\,\} }$$
(P)

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.

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Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3216, USA

    Gábor Pataki

Authors
  1. Gábor Pataki
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Correspondence to Gábor Pataki .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

Additional information

Dedicated to Jonathan Borwein on the occasion of his 60th birthday

Communicated By Heinz H. Bauschke.

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Pataki, G. (2013). Strong Duality in Conic Linear Programming: Facial Reduction and Extended Duals. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_28

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