# 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.

## References

- 1.Auslender, A.: Closedness criteria for the image of a closed set by a linear operator. Numer. Funct. Anal. Optim.
**17**, 503–515 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Barker, G.P., Carlson, D.: Cones of diagonally dominant matrices. Pac. J. Math.
**57**, 15–32 (1975)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Bauschke, H., Borwein, J.M.: Conical open mapping theorems and regularity. In: Proceedings of the Centre for Mathematics and its Applications 36, pp. 1–10. Australian National University (1999)Google Scholar
- 4.Berman, A.: Cones, Matrices and Mathematical Programming. Springer, Berlin (1973)CrossRefzbMATHGoogle Scholar
- 5.Bertsekas, D., Tseng, P.: Set intersection theorems and existence of optimal solutions. Math. Progr.
**110**, 287–314 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
- 7.Bonnans, F.J., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
- 8.Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples. CMS Books in Mathematics. Springer, Berlin (2000)CrossRefGoogle Scholar
- 9.Borwein, J.M., Moors, W.B.: Stability of closedness of convex cones under linear mappings. J. Convex Anal.
**16**(3–4), 699–705 (2009)MathSciNetzbMATHGoogle Scholar - 10.Borwein, J.M., Moors, W.B.: Stability of closedness of convex cones under linear mappings II. J. Nonlinear Anal. Optim.
**1**(1), 1–7 (2010)Google Scholar - 11.Borwein, J.M., Wolkowicz, H.: Facial reduction for a cone-convex programming problem. J. Aust. Math. Soc.
**30**, 369–380 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Borwein, J.M., Wolkowicz, H.: Regularizing the abstract convex program. J. Math. Anal. App.
**83**, 495–530 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Cheung, V., Wolkowicz, H., Schurr, S.: Preprocessing and regularization for degenerate semidefinite programs. In: Bailey, D., Bauschke, H.H., Garvan, F., Théra, M., Vanderwerff, J.D., Wolkowicz, H. (eds.) Proceedings of Jonfest: A Conference in Honour of the 60th Birthday of Jon Borwein. Springer, Berlin (2013)Google Scholar
- 14.Chua, C.B., Tunçel, L.: Invariance and efficiency of convex representations. Math. Progr. B
**111**, 113–140 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Drusviyatsky, D., Pataki, G., Wolkowicz, H.: Coordinate shadows of semi-definite and euclidean distance matrices. SIAM J. Opt.
**25**(2), 1160–1178 (2015)CrossRefzbMATHGoogle Scholar - 16.Güler, O.: Foundations of Optimization. Graduate Texts in Mathematics. Springer, Berlin (2010)Google Scholar
- 17.Glineur, F.: Proving strong duality for geometric optimization using a conic formulation. Ann. Oper. Res.
**105**(2), 155–184 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Gortler, S.J., Thurston, D.P.: Characterizing the universal rigidity of generic frameworks. Discrete Comput. Geom.
**51**(4), 1017–1036 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Klep, I., Schweighofer, M.: An exact duality theory for semidefinite programming based on sums of squares. Math. Oper. Res.
**38**(3), 569–590 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Opt.
**20**, 2679–2708 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Liu, M., Pataki, G.: Exact duality in semidefinite programming based on elementary reformulations. SIAM J. Opt.
**25**(3), 1441–1454 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Lourenco, B., Muramatsu, M., Tsuchiya, T.: Facial reduction and partial polyhedrality. Optimization Online. http://www.optimization-online.org/DB_FILE/2015/11/5224.pdf (2015)
- 23.Lourenco, B., Muramatsu, M., Tsuchiya, T.: A structural geometrical analysis of weakly infeasible SDPs. J. Oper. Res. Soc. Jpn.
**59**(3), 241–257 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Pataki, G.: The geometry of semidefinite programming. In: Saigal, R.,Vandenberghe, L., Wolkowicz, H. (eds.) Handbook of semidefiniteprogramming. Kluwer Academic Publishers. Also available from www.unc.edu/~pataki (2000)
- 25.Pataki, G.: On the closedness of the linear image of a closed convex cone. Math. Oper. Res.
**32**(2), 395–412 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Pataki, G.: On the connection of facially exposed and nice cones. J. Math. Anal. App.
**400**, 211–221 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Pataki, G.: Strong duality in conic linear programming: facialreduction and extended duals. In: Bailey, D., Bauschke, H.H.,Garvan, F., Théra, M., Vanderwerff, J.D., Wolkowicz, H. (eds.) Proceedings of Jonfest: A Conference in Honour of the 60th Birthdayof Jon Borwein. Springer. Also available from http://arxiv.org/abs/1301.7717 (2013)
- 28.Pataki, G.: Bad semidefinite programs: they all look the same. SIAM J. Opt.
**27**(1), 146–172 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Permenter, F., Parrilo, P.: Partial facial reduction: simplified, equivalent sdps via approximations of the psd cone. Technical Report. http://arxiv.org/abs/1408.4685 (2014)
- 30.Pólik, I., Terlaky, T.: Exact duality for optimization over symmetric cones. Lehigh University, Betlehem, PA, USA. Technical Report (2009)Google Scholar
- 31.Provan, J.S., Shier, D.R.: A paradigm for listing (s, t)-cuts in graphs. Algorithmica
**15**(4), 351–372 (1996)MathSciNetzbMATHGoogle Scholar - 32.Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Progr. Ser. B
**77**, 129–162 (1997)MathSciNetzbMATHGoogle Scholar - 33.Ramana, M.V., Freund, R.: On the elsd duality theory for sdp. Technical Report. MIT (1996)Google Scholar
- 34.Ramana, M.V., Tunçel, L., Wolkowicz, H.: Strong duality for semidefinite programming. SIAM J. Opt.
**7**(3), 641–662 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 35.Read, R., Tarjan, R.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks
**5**, 237–252 (1975)MathSciNetCrossRefzbMATHGoogle Scholar - 36.Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia, USA (2001)Google Scholar
- 37.Rockafellar, T.R.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
- 38.Roshchina, V.: Facially exposed cones are not nice in general. SIAM J. Opt.
**24**, 257–268 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 39.Waki, H.: How to generate weakly infeasible semidefinite programs via Lasserre’s relaxations for polynomial optimization. Optim. Lett.
**6**(8), 1883–1896 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 40.Waki, H., Muramatsu, M.: Facial reduction algorithms for conic optimization problems. J. Optim. Theory Appl.
**158**(1), 188–215 (2013)MathSciNetCrossRefzbMATHGoogle Scholar