# Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming

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## Abstract

In conic linear programming—in contrast to linear programming—the Lagrange dual is not an exact dual: it may not attain its optimal value, or there may be a positive duality gap. The corresponding Farkas’ lemma is also not exact (it does not always prove infeasibility). We describe exact duals, and certificates of infeasibility and weak infeasibility for conic LPs which are nearly as simple as the Lagrange dual, but do not rely on any constraint qualification. Some of our exact duals generalize the SDP duals of Ramana, and Klep and Schweighofer to the context of general conic LPs. Some of our infeasibility certificates generalize the row echelon form of a linear system of equations: they consist of a small, trivially infeasible subsystem obtained by elementary row operations. We prove analogous results for weakly infeasible systems. We obtain some fundamental geometric corollaries: an exact characterization of when the linear image of a closed convex cone is closed, and an exact characterization of nice cones. Our infeasibility certificates provide algorithms to generate *all* infeasible conic LPs over several important classes of cones; and *all* weakly infeasible SDPs in a natural class. Using these algorithms we generate a public domain library of infeasible and weakly infeasible SDPs. The status of our instances can be verified by inspection in exact arithmetic, but they turn out to be challenging for commercial and research codes.

## Keywords

Conic linear programming Semidefinite programming Facial reduction Exact duals Exact certificates of infeasibility and weak infeasibility Closedness of the linear image of a closed convex cone## Mathematics Subject Classification

90C46 49N15 90C22 90C25 52A40## Notes

### Acknowledgements

We are grateful to the referees, the Associate Editor, and Melody Zhu for their insightful comments, and to Imre Pólik for his help in our work with the SDP solvers.

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