Abstract
We characterize the smallest (best) barrier parameter of self-concordant barriers for homogeneous convex cones. In particular, we prove that this parameter is the same as the rank of the cone which is the number of steps in a recursive construction of the cone (Siegel domain construction). We also provide lower bounds on the barrier parameter in terms of the Carathéodory number of the cone. The bounds are tight for homogeneous self-dual cones. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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References
J. Dorfmeister, Homogeneous Siegel domains,Nagoya Math. J. 86 (1982) 39–83.
J. Faraut and A. Korányi,Analysis on Symmetric Cones (Oxford University Press, New York, 1994).
L. Faybusovich, Jordan algebras, symmetric cones and interior-point methods, Manuscript, Dept. of Mathematics, University of Notre Dame, IN, USA, 1995.
S.G. Gindikin, Analysis in homogeneous domains,Russian Math. Surveys 19 (1964) 1–89.
S.G. Gindikin,Tube domains and the Cauchy problem, Translation of Mathematical Monographs, Vol. 111 (AMS Providence, RI, 1992).
O. Güler, Barrier functions in interior point methods,Mathematics of Operations Research 21 (1996) 860–885.
O. Güler and L. Tunçel, Invariant barrier functions on homogeneous convex cones,in preparation.
A. Krieg,Modular Forms on Half-Spaces of Quaternions, Lecture Notes in Mathematics, Vol. 1143 (Springer, New York, 1985).
A.S. Nemirovskii, Private communication, 1995.
Yu.E. Nesterov and A.S. Nemirovskii,Interior-Point Polynomial Algorithms in Convex Programming (SIAM Publications, Philadelphia, PA, 1994).
Yu.E. Nesterov and M.J. Todd, Self-scaled barriers and interior-point methods for convex programming,Mathematics of Operations Research 22 (1997) 1–46.
Yu.E. Nesterov and M.J. Todd, Primal-dual interior-point methods for self-scaled cones, SIAM Journal on Optimization (1997), to appear.
M. Ramana, L. Tunçel and H. Wolkowicz, Strong duality for semidefinite programming,SIAM Journal on Optimization (1997), to appear.
O.S. Rothaus, Domains of Positivity.Abh. Math. Sem. Univ. Hamburg 24 (1960) 189–235.
O.S. Rothaus, The construction of homogeneous convex cones,Annals of Math. 83 (1966) 358–376.
R. Schneider,Convex Bodies: The Brunn—Minkowski Theory (Cambridge University Press, Cambridge, UK, 1993).
È.B. Vinberg, The theory of homogeneous cones,Trans. Moscow Math. Soc. 12 (1965) 340–403.
È.B. Vinberg, The structure of the group of automorphisms of a homogeneous convex cone,Trans. Moscow Math. Soc. 13 (1965) 63–93.
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Research supported in part by an operating grant from NSERC of Canada.
Research supported in part by the National Science Foundation under grant DMS-9306318.
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Güler, O., Tunçel, L. Characterization of the barrier parameter of homogeneous convex cones. Mathematical Programming 81, 55–76 (1998). https://doi.org/10.1007/BF01584844
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DOI: https://doi.org/10.1007/BF01584844