Skip to main content
SpringerLink
Log in

Conditioning of Random Conic Systems Under a General Family of Input Distributions

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

We consider the conic feasibility problem associated with the linear homogeneous system Ax≤0, x≠0. The complexity of iterative algorithms for solving this problem depends on a condition number C(A). When studying the typical behavior of algorithms under stochastic input, one is therefore naturally led to investigate the fatness of the tails of the distribution of C(A). Introducing the very general class of uniformly absolutely continuous probability models for the random matrix A, we show that the distribution tails of C(A) decrease at algebraic rates, both for the Goffin–Cheung–Cucker number C G and the Renegar number C R . The exponent that drives the decay arises naturally in the theory of uniform absolute continuity, which we also develop in this paper. In the case of C G , we also discuss lower bounds on the tail probabilities and show that there exist absolutely continuous input models for which the tail decay is subalgebraic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K.M. Anstreicher, J. Ji, F.A. Potra, Y. Ye, Probabilistic analysis of an infeasible-interior-point algorithm for linear programming, Math. Oper. Res. 24(1), 176–192 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Google Scholar 

  4. K.H. Borgwardt, 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 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Cheung, F. Cucker, A new condition number for linear programming, Math. Program. 91, 163–174 (2001).

    MATH  MathSciNet  Google Scholar 

  6. D. Cheung, F. Cucker, R. Hauser, Tail decay and moment estimates of a condition number for random linear conic systems, SIAM J. Optim. 15(4), 1237–1261 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Cucker, J. Peña, A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine, SIAM J. Optim. 12, 522–554 (2002).

    Article  Google Scholar 

  8. F. Cucker, M. Wschebor, On the expected condition number of linear programming problems, Numer. Math. 94, 419–478 (2002).

    Article  MathSciNet  Google Scholar 

  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).

    MATH  Google Scholar 

  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, The relaxation method for solving systems of linear inequalities, Math. Oper. Res. 5, 388–414 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Google Scholar 

  16. P. Huhn, K.H. Borgwardt, Interior-point methods: worst case and average case analysis of a phase-I algorithm and a termination procedure, J. Complex. 18(3), 833–910 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Renegar, Linear programming, complexity theory and elementary functional analysis, Math. Program. 70, 279–351 (1995).

    MathSciNet  Google Scholar 

  18. R. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms (Spartan Books, Washington, 1962).

    MATH  Google Scholar 

  19. S.M. Ross, Probability Models in Computer Science (Harcourt/Academic Press, San Diego, 2002).

    Google Scholar 

  20. S. Smale, On the average number of steps of the simplex method of linear programming, Math. Program. 27, 241–262 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  21. D. Spielman, S. Teng, Smoothed analysis: Why the simplex algorithm usually takes polynomial time, J. Assoc. Comput. Mach. 51, 385–463 (2004).

    MathSciNet  Google Scholar 

  22. M.J. Todd, L. Tunçel, Y. Ye, Characterizations, bounds and probabilistic analysis of two complexity measures for linear programming problems, Math. Program. 90, 59–69 (2001).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK

    Raphael Hauser

  2. EURANDOM, Technische Universiteit Eindhoven, P.O. Box 513, 5600, MB, Eindhoven, The Netherlands

    Tobias Müller

Authors
  1. Raphael Hauser
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raphael Hauser.

Additional information

Communicated by Felipe Cucker and Mike Todd.

R. Hauser was supported by a grant of the Nuffield Foundation under the “Newly Appointed Lecturers” grant scheme (project number NAL/00720/G) and through grant GR/S34472 from the Engineering and Physical Sciences Research Council of the UK.

The research in this paper was conducted while T. Müller was a research student at the University of Oxford. He was partially supported by EPSRC, the Oxford University Department of Statistics, Bekker-la-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hauser, R., Müller, T. Conditioning of Random Conic Systems Under a General Family of Input Distributions. Found Comput Math 9, 335–358 (2009). https://doi.org/10.1007/s10208-008-9034-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-008-9034-0

Keywords

AMS Subject Classifications

Search

Navigation