Conditioning of random conic systems under a general family of input distributions

Foundations of Computational Mathematics

Volumn 9, Issue 3, 2009, Pages 335-358

  Hauser, Raphael ^a   Müller, Tobias ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Condition numbers; Conic feasibility problem; Linear programming; Probabilistic analysis; Random matrices; Smoothed analysis

Indexed keywords

CONDITION NUMBERS; FEASIBILITY PROBLEM; PROBABILISTIC ANALYSIS; RANDOM MATRICES; SMOOTHED ANALYSIS;

ALGORITHMS; LINEAR PROGRAMMING; MATRIX ALGEBRA; NUMBER THEORY; PROBABILITY DISTRIBUTIONS; PROBLEM SOLVING;

C (PROGRAMMING LANGUAGE);

EID: 64749097879     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-008-9034-0     Document Type: Article

References (22)

1. K.M. Anstreicher J. Ji F.A. Potra Y. Ye 1999 Probabilistic analysis of an infeasible-interior-point algorithm for linear programming Math. Oper. Res. 24 1 176 192
2. A. Blum, J. Dunagan, Smoothed analysis of the perceptron algorithm for linear programming, in SODA'02, pp. 905-914 (2002).
3. L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, Berlin, 1998).
4. K.H. Borgwardt 1982 Some distribution-independent results about the asymptotic order of the average number of pivot steps of the simplex method Math. Oper. Res. 7 441 462
5. D. Cheung F. Cucker 2001 A new condition number for linear programming Math. Program. 91 163 174
6. D. Cheung F. Cucker R. Hauser 2005 Tail decay and moment estimates of a condition number for random linear conic systems SIAM J. Optim. 15 4 1237 1261
7. F. Cucker J. Peña 2002 A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine SIAM J. Optim. 12 522 554
8. F. Cucker M. Wschebor 2002 On the expected condition number of linear programming problems Numer. Math. 94 419 478
9. J. Dunagan, D.A. Spielman, S.-H. Teng, Smoothed analysis of Renegar's condition number for linear programming. Preprint http://theory.lcs.mit.edu/ ~spielman (2003).
10. J. Dunagan, S. Vempala, A simple polynomial-time rescaling algorithm for solving linear programs, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 315-320 (2004).
11. H. Federer, Geometric Measure Theory (Springer, Berlin, 1969).
12. J.-L. Goffin, On the finite convergence of the relaxation method for solving systems of inequalities. Ph.D. thesis, University of California, Berkeley (1971).
13. J.-L. Goffin 1980 The relaxation method for solving systems of linear inequalities Math. Oper. Res. 5 388 414
14. R. Hauser, T. Müller, Algebraic tail decay of condition numbers for random conic systems under a general family of input distributions. Numerical Analysis Group Research Report NA-06/01, Oxford University Computing Laboratory (2006).
15. P. Huhn, K.H. Borgwardt, An upper bound for the average number of iterations required in phase II of an interior-point-method, in Operations Research Proceedings 1997 (Jena) (Springer, Berlin, 1998), pp. 19-24.
16. P. Huhn K.H. Borgwardt 2002 Interior-point methods: worst case and average case analysis of a phase-I algorithm and a termination procedure J. Complex. 18 3 833 910
17. J. Renegar 1995 Linear programming, complexity theory and elementary functional analysis Math. Program. 70 279 351
18. R. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms (Spartan Books, Washington, 1962).
19. S.M. Ross, Probability Models in Computer Science (Harcourt/Academic Press, San Diego, 2002).
20. S. Smale 1983 On the average number of steps of the simplex method of linear programming Math. Program. 27 241 262
21. D. Spielman S. Teng 2004 Smoothed analysis: Why the simplex algorithm usually takes polynomial time J. Assoc. Comput. Mach. 51 385 463
22. M.J. Todd L. Tunçel Y. Ye 2001 Characterizations, bounds and probabilistic analysis of two complexity measures for linear programming problems Math. Program. 90 59 69