Advertisement

Mathematical Programming

, Volume 118, Issue 2, pp 279–299 | Cite as

Projective re-normalization for improving the behavior of a homogeneous conic linear system

  • Alexandre Belloni
  • Robert M. FreundEmail author
FULL LENGTH PAPER

Abstract

In this paper we study the homogeneous conic system \(F: Ax = 0, x \in C\setminus \{0\}\) . We choose a point \(\bar s \in {\rm int}C^*\) that serves as a normalizer and consider computational properties of the normalized system \(F_{\bar s} : Ax = 0, \bar s^T x = 1, x \in C\) . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value \(\vartheta\) of the barrier for C and on the symmetry of the origin in the image set \(H_{\bar s} := \{Ax {\bar s}^Tx = 1, x \in C\}\) , where the symmetry of 0 in \(H_{\bar s}\) is
$$ {\rm sym}(0, H_{\bar s}) := {\rm max}\{\alpha : y \in H_{\bar s} \Rightarrow -\alpha y \in H_{\bar s} \} .$$
We show that a solution of F can be computed in \(O(\sqrt{\vartheta}{\rm ln}(\vartheta/{\rm sym}(0, H_{\bar s}))\) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region \(F_{\bar s}\) and the image set \(H_{\bar s}\) and prove the existence of a normalizer \({\bar s}\) such that \({\rm sym}(0,H_{\bar s}) \ge 1/m\) provided that F has an interior solution. We develop a methodology for constructing a normalizer \({\bar s}\) such that \({\rm sym}(0, H_{\bar s}) \ge 1/m\) with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomial-time, the normalizer will yield a conic system that is solvable in \(O(\sqrt{\vartheta}{\rm ln}(m\vartheta))\) iterations, which is strongly-polynomial-time. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1,000  ×  5,000.

Mathematics Subject Classification (2000)

90C05 90C25 90C51 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belloni, A., Freund, R.M.: Proejctive pre-conditioners for improving the behavior of a homogeneous conic linear system. Working Paper OR375-05, MIT, Operations Research Center (2005)Google Scholar
  2. 2.
    Belloni, A., Freund, R.M.: On the symmetry function of a convex set. Math. Program. (2006, to appear)Google Scholar
  3. 3.
    Bertsimas D. and Vempala S. (2004). Solving convex programs by random walks. J. ACM 51(4): 540–556 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Dyer M.E. and Frieze A.M. (1988). On the complexity of computing the volume of a polyhedron. SIAM J. Comput. Arch. 17(5): 967–974 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Epelman M. and Freund R.M. (2002). A new condition measure, preconditioners and relations between different measures of conditioning for conic linear systems. SIAM J. Optim. 12(3): 627–655 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fishman G.S. (1994). Choosing sample path length and number of sample paths when starting at steady state. Oper. Res. Lett. 16(4): 209–220 zbMATHCrossRefGoogle Scholar
  7. 7.
    Freund R.M. (1989). Combinatorial analogs of Brouwer’s fixed-point theorem on a bounded polyhedron. J. Comb. Theory Ser. B 47(2): 192–219 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Freund R.M. (1991). Projective transformation for interior-point algorithms and a superlinearly convergent algorithm for the w-center problem. Math. Program. 58: 203–222 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Geyer C.J. (1992). Practical markov chain monte carlo. Stat. Sci. 7(4): 473–511 CrossRefMathSciNetGoogle Scholar
  10. 10.
    Grötschel M., Lovász L. and Schrijver A. (1994). Geometric Algorithms and Combiantorial Optimization. Springer, Berlin Google Scholar
  11. 11.
    Grünbaum B. (1967). Convex Polytopes. Wiley, New York zbMATHGoogle Scholar
  12. 12.
    Hammer P.C. (1951). The centroid of a convex body. Proc. Am. Math. Soc. 5: 522–525 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Kalai A. and Vempala S. (2006). Simulating annealing for convex optimization. Math. Oper. Res. 31(2): 253–266 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kelner, J.A., Spielman, D.A.: A randomized polynomial-time simplex algorithm for linear programming. Technical report. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (2006)Google Scholar
  15. 15.
    Khachiyan L. (1996). Rounding of polytopes in the real number model of computation. Math. Oper. Res. 21(2): 307–320 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Leindler L. (1972). On a certain converse of Hölder’s inequality ii. Acta Sci. Math. Szeged 33: 217–223 zbMATHMathSciNetGoogle Scholar
  17. 17.
    Lovász, L., Vempala, S.: The geometry of logconcave functions and an O *(n 3) sampling algorithm. Microsoft Technical ReportGoogle Scholar
  18. 18.
    Lovász, L., Vempala, S.: Hit-and-run is fast and fun. Microsoft Technical ReportGoogle Scholar
  19. 19.
    Lovász, L., Vempala, S.: Where to start a geometric walk? Microsoft Technical ReportGoogle Scholar
  20. 20.
    Minkowski H. (1911). Allegemeine lehzätze über konvexe polyeder. Ges. Abh. Leipzog-Berlin 1: 103–121 Google Scholar
  21. 21.
    Nesterov Y. and Nemirovskii A. (1993). Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia Google Scholar
  22. 22.
    Nesterov Y. and Nemirovskii A. (2003). Central path and Riemannian distances. Université Catholique de Louvain, Belgium, Technical report, CORE Discussion Paper, COREGoogle Scholar
  23. 23.
    Nesterov Y., Todd M.J. and Ye Y. (1999). Infeasible-start primal-dual methods and infeasibility detectors. Math. Program. 84: 227–267 zbMATHMathSciNetGoogle Scholar
  24. 24.
    Prékopa A. (1973). Logarithmic concave measures and functions. Acta Sci. Math. Szeged 34: 335–343 zbMATHMathSciNetGoogle Scholar
  25. 25.
    Prékopa A. (1973). On logarithmic concave measures with applications to stochastic programming. Acta Sci. Math. Szeged 32: 301–316 Google Scholar
  26. 26.
    Renegar J. A (2001). Mathematical View of Interior-Point Methods in Convex Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia Google Scholar
  27. 27.
    Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton zbMATHGoogle Scholar
  28. 28.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: SDPT3—a MATLAB software package for semidefinite-quadratic-linear programming, version 3.0. Technical report, Available at http://www.math.nus.edu.sg/~mattohkc/sdpt3.html (2001)
  29. 29.
    Zhang Y. and Gao L. (2003). On numerical solution of the maximum volume ellipsoid problem. SIAM J. Optim. 14(1): 53–76 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Fuqua School of BusinessDuke UniversityDurhamUSA
  2. 2.MIT Sloan School of ManagementCambridgeUSA

Personalised recommendations