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

Abstract

The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear programin 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, 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 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.Dedicated to Jonathan Borwein on the occasion of his 60th birthday

A Structural Geometrical Analysis of Weakly Infeasible SDPS

“…We now take a closer look at the connection between FRA/CEA and FP. More details on FRA can be found in [15] and [26]. In [26], it is also explained the connection between FRA and CEA, see section 4 therein.…”

“…ELSD has the remarkable feature that the size of the extended problem is bounded by a polynomial in terms of the size of the original problem. In [22], Ramana, Tunçel and Wolkowicz clarified the connection between ELSD and facial reduction, see also [15]. In [17], Pólik and Terlaky provided strong duals for conic programming over symmetric cones.…”

“…See the excellent article by Pataki [15] for an elementary exposition of FRA, where he points out the relation between facial reduction and extended duals. Pataki also found that all "badly behaved" semidefinite programs can be reduced to a common 2 × 2 semidefinite program [16].…”

“…Regarding finite certificates, it is worthwhile mentioning that there is a way to obtain a finite and polynomially bounded certificate of weak infeasibility by using Ramana's extended Lagrangian dual [21] or the facial reduction algorithm [3,4,15,26]. This result is based on the fact that (K n , L, c) is weakly infeasible if and only if it is infeasible and not strongly infeasible.…”

Linear and Nonlinear Semidefinite Programming

“…Another general approach to deal with the lack of strict feasibility bases on the so called facial reduction and extended duals [24,25,26,122,123,135,115]. Let us discuss this second approach, since it uses geometric properties of the semidefinite cone and provides in general smaller regularized problems.…”