Deterministic and stochastic methods for computing volumetric moduli of convex cones

Computational and Applied Mathematics

Volumn 29, Issue 2, 2010, Pages 215-246

  Gourion, Daniel ^a   Seeger, Alberto ^a  

^a UNIVERSITY OF AVIGNON   (France)

Author keywords

Convex cone; Randomness; Solid angle; Volumetric modulus

Indexed keywords

CONES;

CONVEX CONE; DETERMINISTIC METHODS; DETERMINISTIC TECHNIQUE; EUCLIDEAN SPACES; RANDOMNESS; SOLID ANGLE; STOCHASTIC METHODS; STOCHASTIC TECHNIQUES; VOLUMETRIC MODULUS; VOLUMETRICS;

STOCHASTIC SYSTEMS;

EID: 77956554596     PISSN: 22383603     EISSN: 18070302     Source Type: Journal    
DOI: 10.1590/S1807-03022010000200007     Document Type: Article

References (24)

1. B. Alkire and L. Vandenberghe, Convex optimization problems involving finite autocorrelation sequences. Math. Program., 93 (2002), 331-359.
2. V. Boyarshinov and M. Magdon-Ismail, Linear time isotonic and unimodal regression in the L1 and L∞ norms. J. Discrete Algorithms, 4 (2006), 676-691.
3. B. Dumitrescu, P. Stoica and I. Tǎbuş, On the parameterization of positive real sequences and MA parameter estimation. IEEE Trans. Signal Process., 49 (2001), 2630-2639.
4. R.L. Dykstra and J.H. Lemke, Duality of I projections and maximum likelihood estimation for log-linear models under cone constraints. J. Amer. Statist. Assoc., 83 (1988), 546-554.
5. M.L. Eaton and T. Kariya, Robust tests for spherical symmetry. Ann. Statist., 5 (1977), 206-215.
6. M. Epelman and R.M. Freund, A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systems. SIAM J. Optim., 12 (2002), 627-655.
7. F. Eriksson, On the measure of solid angles. Math. Mag., 63 (1990), 184-187.
8. R.M. Freund, On the primal-dual geometry of level sets in linear and conic optimization. SIAM J. Optim., 13 (2003), 1004-1013.
9. R.M. Freund and J.R. Vera, Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm. SIAM J. Optim., 10 (1999), 155-176.
10. M.P.C. Fossorier and A. Valembois, Sphere-packing bounds revisited for moderate block lengths. IEEE Trans. Inform. Theory, 50 (2004), 2998-3014.
11. W. Gao and N.Z. Shi, I-projection onto isotonic cones and its applications to maximum likelihood estimation for log-linear models. Ann. Inst. Statist. Math., 55 (2003), 251-263.
12. D. Gourion and A. Seeger, Critical angles in polyhedral convex cones: numerical and statistical considerations. Math. Program., 123 (2010), 173-198.
13. R. Henrion and A. Seeger, On properties of different notions of centers for convex cone. Set-Valued Anal., 18 (2010), 205-231.
14. R. Henrion and A. Seeger, Inradius and circumradius of various convex cones arising in applications. Set-Valued Anal., to appear.
15. A. Iusem and A. Seeger, Axiomatization of the index of pointedness for closed convex cones. Comput. Applied Math., 24 (2005), 245-283.
16. A. Iusem and A. Seeger, Measuring the degree of pointedness of a closed convex cone: a metric approach. Math. Nachr., 279 (2006), 599-618.
17. A. Iusem and A. Seeger, Normality and modulability indices. Part I: convex cones in normed spaces. J. Math. Anal. Appl., 338 (2008), 392-406.
18. A. Iusem and A. Seeger, Antipodality in convex cones and distance to unpointedness. Appl. Math. Letters, 21 (2008), 1018-1023.
19. A. Iusem and A. Seeger, Antipodal pairs, critical pairs, and Nash angular equilibria in convex cones. Optimiz. Meth. Software, 23 (2008), 73-93.
20. J.M. Ribando, Measuring solid angles beyond dimension three. Discrete Comput. Geom., 36 (2006), 479-487.
21. A. van Oosterom and J. Strackee, The solid angle of a plane triangle. IEEE Trans. Biomed. Eng., 30 (1983), 125-126.
22. A. Seeger and M. Torki, Local minima of quadratic forms on convex cones. J. Global Optim., 44 (2009), 1-28.
23. C. Shannon, Probability of error for optimal codes in a Gaussian channel. Bell System Tech. J., 38 (1959), 611-656.
24. R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., 576 (1989), 500-535.