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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.

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Correspondence to Gábor Pataki .

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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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