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Mathematical Programming

, Volume 119, Issue 1, pp 95–107 | Cite as

A geometric analysis of Renegar’s condition number, and its interplay with conic curvature

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

Abstract

For a conic linear system of the form AxK, K a convex cone, several condition measures have been extensively studied in the last dozen years. Among these, Renegar’s condition number \({\mathcal{C}}(A)\) is arguably the most prominent for its relation to data perturbation, error bounds, problem geometry, and computational complexity of algorithms. Nonetheless, \({\mathcal{C}}(A)\) is a representation-dependent measure which is usually difficult to interpret and may lead to overly conservative bounds of computational complexity and/or geometric quantities associated with the set of feasible solutions. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K; furthermore our bounds highlight the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar’s condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements the literature where only lower bounds have heretofore been developed.

Mathematics Subject Classification (2000)

90C31 52A20 

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References

  1. 1.
    Belloni, A., Freund, R.M., Vempala, S.: An efficient re-scaled perceptron algorithm for conic systems. MIT Operations Research Center Working Paper (2006), no. OR-379-06Google Scholar
  2. 2.
    Bertsekas D. (1999). Nonlinear Programming. Athena Scientific, Nashua zbMATHGoogle Scholar
  3. 3.
    Freund R.M. and Vera J.R. (1999). Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm. SIAM J. Optimi. 10(1): 155–176 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Freund R.M. and Vera J.R. (1999). Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system. Math. Program. 86(2): 225–260 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Freund R.M. (2004). Complexity of convex optimization using geometry-based measures and a reference point. Math. Program. 99: 197–221 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Levitin E. and Poljak B.J. (1963). Constrained minimization methods. U.S.S.R. Comput. Math. Math. Phys. 6: 787–823 MathSciNetGoogle Scholar
  7. 7.
    Poljak B.J. (1966). Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. 7: 72–75 zbMATHGoogle Scholar
  8. 8.
    Renegar J. (1994). Some perturbation theory for linear programming. Math. Program. 65(1): 73–91 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Renegar J. (1995). Linear programming, complexity theory and elementary functional analysis. Math. Program. 70(3): 279–351 CrossRefMathSciNetGoogle Scholar
  10. 10.
    Vial J.-P. (1982). Strong convexity of sets and functions. J. Math. Econ. 9: 187–205 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Vial J.-P. (1983). Strong and weak convexity of sets and functions. Math. Oper. Res. 8(2): 231–259 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

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

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