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

TL;DR

This paper clarifies facial reduction and extended duals in conic linear programming, generalizes Ramana's dual for nice cones, and provides a simple, correct facial reduction algorithm applicable to a broad class of cones.

Contribution

It offers a clear exposition of facial reduction, proves a simple algorithm's correctness, and constructs a family of extended duals for nice cones, broadening the applicability.

Findings

Facial reduction algorithm is simple and correct.

Extended duals are constructed for nice cones, including SDP.

The approach generalizes Ramana's dual to a wider class of cones.

Abstract

The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program $$ (P) \sup {<c, x> | Ax \leq_K b} $$ 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.