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Approximate cone factorizations and lifts of polytopes. (English) Zbl 1319.52010
Summary: In this paper, we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix. This provides a robust generalization of the famous result of M. Yannakakis [J. Comput. Syst. Sci. 43, No. 3, 441–466 (1991; Zbl 0748.90074)] that polyhedral lifts of a polytope are controlled by (exact) nonnegative factorizations of its slack matrix. Our approximations behave well under polarity and have efficient representations using second order cones. We establish a direct relationship between the quality of the factorization and the quality of the approximations, and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron.

52A23 Asymptotic theory of convex bodies
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
Full Text: DOI arXiv
[1] Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3-51 (2003) · Zbl 1153.90522
[2] Borwein, J; Wolkowicz, H, Regularizing the abstract convex program, J. Math. Anal. Appl., 83, 495-530, (1981) · Zbl 0467.90076
[3] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) · Zbl 1058.90049
[4] Braun, G., Fiorini, S., Pokutta, S., Steurer, D.: Approximation limits of linear programs (beyond hierarchies). In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 480-489. IEEE (2012) · Zbl 1343.68308
[5] Braverman, M., Moitra, A.: An information complexity approach to extended formulations. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 19, (2012) · Zbl 1293.68137
[6] Gillis, N; Glineur, F, On the geometric interpretation of the nonnegative rank, Linear Algebra Its Appl., 437, 2685-2712, (2012) · Zbl 1258.65039
[7] Gouveia, J; Parrilo, PA; Thomas, RR, Lifts of convex sets and cone factorizations, Math. Oper. Res., 38, 248-264, (2013) · Zbl 1291.90172
[8] Gouveia, J., Robinson, R. Z., Thomas, R. R.: Worst-Case Results for Positive Semidefinite Rank. arXiv:1305.4600 (2013) · Zbl 1344.90046
[9] Lobo, M; Vandenberghe, L; Boyd, S; Lebret, H, Applications of second-order cone programming, Linear Algebra Its Appl., 284, 193-228, (1998) · Zbl 0946.90050
[10] Nesterov, Y.E., Nemirovski, A.: Interior Point Polynomial Methods in Convex Programming, volume 13 of Studies in Applied Mathematics. Siam, Philadelphia (1994)
[11] Pashkovich, K.: Extended Formulations for Combinatorial Polytopes. PhD thesis, Magdeburg Universität (2012) · Zbl 1267.90098
[12] Pataki, G, On the closedness of the linear image of a closed convex cone, Math. Oper. Res., 32, 395-412, (2007) · Zbl 1341.90146
[13] Pataki, G, On the connection of facially exposed and Nice cones, J. Math. Anal. Appl., 400, 211-221, (2013) · Zbl 1267.90098
[14] Renegar, J, Hyperbolic programs, and their derivative relaxations, Found. Comput. Math., 6, 59-79, (2006) · Zbl 1130.90363
[15] Rockafellar, R.T.: Convex Analysis. Princeton Mathematical, vol. 28. Princeton University Press, Princeton (1970) · Zbl 0193.18401
[16] Saunderson, J., Parrilo, P.A.: Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones. Math. Program. Ser. A. (2014). doi:10.1007/s10107-014-0804-y · Zbl 1327.90180
[17] Sonnevend, Gy.: An “analytical centre” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: System Modelling and Optimization, pp. 866-875. Springer, Berlin (1986) · Zbl 0602.90106
[18] Sturm, JF; Zhang, S, An \({O(\sqrt{n L})}\) iteration bound primal-dual cone affine scaling algorithm for linear programming, Math. Program., 72, 177-194, (1996) · Zbl 0853.90085
[19] Yannakakis, M, Expressing combinatorial optimization problems by linear programs, J. Comput. Syst. Sci., 43, 441-466, (1991) · Zbl 0748.90074
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