Coverage processes on spheres and condition numbers for linear programming

Annals of Probability

Volumn 38, Issue 2, 2010, Pages 570-604

  Bürgisser, Peter ^a   Cucker, Felipe ^b   Lotz, Martin ^c  

^a UNIVERSITY OF PADERBORN   (Germany)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Condition numbers; Covering processes; Geometric probability; Integral geometry; Linear programming.

Indexed keywords

EID: 77953646724     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOP489     Document Type: Article

References (31)

1. AGMON, S. (1954). The relaxation method for linear inequalities. Canad. J. Math. 6 382-392. MR0062786
2. BÜRGISSER, P. and AMELUNXEN, D. (2008). Uniform smoothed analysis of a condition number for linear programming. Accepted for Math. Program. A. Available at arXiv:0803.0925.
3. CHEUNG, D. and CUCKER, F. (2001). A new condition number for linear programming. Math. Program. 91 163-174. MR1865268
4. CHEUNG, D. and CUCKER, F. (2002). Probabilistic analysis of condition numbers for linear programming. J. Optim. Theory Appl. 114 55-67. MR1910854
5. CHEUNG, D., CUCKER, F. and HAUSER, R. (2005). Tail decay and moment estimates of a condition number for random linear conic systems. SIAM J. Optim. 15 1237-1261. MR2178497
6. CUCKER, F. and PEÑA, J. (2002). A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine. SIAM J. Optim. 12 522-554. MR1885574
7. CUCKER, F. and WSCHEBOR, M. (2002). On the expected condition number of linear programming problems. Numer. Math. 94 419-478. MR1981163
8. DUNAGAN, J., SPIELMAN, D. A. and TENG, S.-H. (2009). Smoothed analysis of condition numbers and complexity implications for linear programming. Math. Programming. To appear. Available at http://arxiv.org/abs/cs/0302011v2.
9. DVORETZKY, A. (1956). On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. USA 42 199-203. MR0079365
10. GILBERT, E. N. (1965). The probability of covering a sphere with N circular caps. Biometrika 52 323-330. MR0207005
11. GOFFIN, J.-L. (1980). The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5 388-414. MR594854
12. HALL, P. (1985). On the coverage of k-dimensional space by k-dimensional spheres. Ann. Probab. 13 991-1002. MR799434
13. HALL, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York. MR973404
14. HAUSER, R. andMÜLLER, T. (2009). Conditioning of random conic systems under a general family of input distributions. Found. Comput. Math. 9 335-358. MR2496555
15. JANSON, S. (1986). Random coverings in several dimensions. Acta Math. 156 83-118. MR822331
16. KAHANE, J.-P. (1959). Sur le recouvrement d'un cercle par des arcs disposés au hasard. C. R. Math. Acad. Sci. Paris 248 184-186. MR0103533
17. MILES, R. E. (1968). Random caps on a sphere. Ann. Math. Statist. 39 1371.
18. MILES, R. E. (1969). The asymptotic values of certain coverage probabilities. Biometrika 56 661-680. MR0254953
19. MILES, R. E. (1971). Isotropic random simplices. Adv. in Appl. Probab. 3 353-382. MR0309164
20. MORAN, P. A. P. and FAZEKAS DE ST. GROTH, S. (1962). Random circles on a sphere. Biometrika 49 389-396. MR0156434
21. MOTZKIN, T. S. and SCHOENBERG, I. J. (1954). The relaxation method for linear inequalities. Canad. J. Math. 6 393-404. MR0062787
22. REITZNER, M. (2002). Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc. 354 2243-2278. MR1885651
23. ROSENBLATT, F. (1962). Principles of Neurodynamics. Perceptrons and the Theory of Brain Mechanisms. Spartan Books, Washington, DC. MR0135635
24. SANTALÓ, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA. MR0433364
25. SIEGEL, A. F. (1979). Asymptotic coverage distributions on the circle. Ann. Probab. 7 651- 661. MR537212
26. SIEGEL, A. F. and HOLST, L. (1982). Covering the circle with random arcs of random sizes. J. Appl. Probab. 19 373-381. MR649974
27. SOLOMON, H. (1978). Geometric Probability. SIAM, Philadelphia, PA. MR0488215
28. STEVENS, W. L. (1939). Solution to a geometrical problem in probability. Ann. Eugenics 9 315-320. MR0001479
29. WENDEL, J. G. (1962). A problem in geometric probability. Math. Scand. 11 109-111. MR0146858
30. WHITWORTH, W. A. (1965). DCC Exercises in Choice and Chance. Dover, New York.
31. ZÄHLE, M. (1990). A kinematic formula and moment measures of random sets. Math. Nachr. 149 325-340. MR1124814