Strong duality in conic linear programming: facial reduction and extended duals [article]

Gabor Pataki
2013 arXiv   pre-print
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program (P) | Ax ≤_K b in the absence of any constraint qualification. The facial reduction algorithm 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, Tuncel, and Wolkowicz showed that these approaches are closely related; in
more » ... ar, they proved the correctness of Ramana's dual using certificates from a facial reduction algorithm. Here we give a clear and self-contained exposition of facial reduction, of extended duals, and generalize Ramana's dual: -- we state a simple facial reduction algorithm and prove its correctness; and -- 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.
arXiv:1301.7717v3 fatcat:x6gca7skjrbfde5ogghz7atm7y