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

Gabor Pataki(gabor***at***unc.edu)

Abstract: The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program (P) \sup { | Ax \leq_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 particular, 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 {\em nice} cone. This class of cones includes the semidefinite cone and other important cones.

Keywords: Conic linear programming; minimal cone; semidefinite progr amming; facial reduction; extended duals; nice cones

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Convex and Nonsmooth Optimization (Convex Optimization )

Citation:

Download: [PDF]

Entry Submitted: 02/13/2013
Entry Accepted: 02/13/2013
Entry Last Modified: 02/13/2013

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