Unifying Condition Numbers for Linear Programming

References

  • Cheung D., Cucker F. A new condition number for linear programming. Math. Programming (2001) 91:163–174CrossrefGoogle Scholar
  • Cheung D., Cucker F. Solving linear programs with finite precision: I. Condition numbers and random programs. Math. Programming (2002) . ForthcomingGoogle Scholar
  • Cheung D., Cucker F. Solving linear programs with finite precision: II. Algorithms. (2003) . Working paperGoogle Scholar
  • Cheung D., Cucker F., Ye Y., Ciarlet P., Cucker F. Linear programming and condition numbers under the real number computation model. Handbook of Numerical Analysis (2003) XI(North-Holland, Amsterdam, The Netherlands) CrossrefGoogle Scholar
  • Cucker F., Peña J. A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine. SIAM J. Optim. (2002) 12:522–554CrossrefGoogle Scholar
  • Demmel J. The componentwise distance to the nearest singular matrix. SIAM J. Matrix Anal. Appl. (1992) 13:10–19CrossrefGoogle Scholar
  • Dontchev A., Lewis A., Rockafellar T. The radius of metric regularity. Trans. Amer. Math. Soc. (2003) 355:493–517CrossrefGoogle Scholar
  • Doyle J. Analysis of feedback systems with structured uncertainty. IEE Proc. (1982) 129:242–250CrossrefGoogle Scholar
  • Eckart C., Young G. The approximation of one matrix by another of lower rank. Psychometrika (1936) 1:211–218CrossrefGoogle Scholar
  • Epelman M., Freund R. M. Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system. Math. Programming (2000) 88:451–485CrossrefGoogle Scholar
  • Epelman M., Freund R. M. A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systems. SIAM J. Optim. (2002) 12:627–655CrossrefGoogle Scholar
  • Freund R. M., Vera J. R. Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm. SIAM J. Optim. (1999a) 10:155–176CrossrefGoogle Scholar
  • Freund R. M., Vera J. R. Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system. Math. Programming (1999b) 86:225–260CrossrefGoogle Scholar
  • Goldman A., Tucker A., Kuhn H., Tucker A. Theory of linear programming. Linear Inequalities and Related Systems, Vol. 38. Ann. Math. Stud. (1956) (Princeton University Press, Princeton, New Jersey) 53–97Google Scholar
  • Higham N.Accuracy and Stability of Numerical Algorithms (1996) (SIAM, Philadelphia PA) Google Scholar
  • Lewis A. Ill-conditioned convex processes and linear inequalities. Math. Oper. Res. (1999) 24:829–834LinkGoogle Scholar
  • Lewis A. S. Ill-conditioned inclusions. Set-Valued Anal. (2001) 9:375–381CrossrefGoogle Scholar
  • Nunez M., Freund R. M. Condition measures and properties of the central trajectory of a linear program. Math. Programming (1998) 83:1–28CrossrefGoogle Scholar
  • Packard A., Doyle J. The complex structured singular value. Automatica (1993) 29:71–109CrossrefGoogle Scholar
  • Packard A., Pandey P. Continuity properties of the real/complex structured singular value. IEEE Trans. Automatic Control (1993) 38:415–428CrossrefGoogle Scholar
  • Peña J. Understanding the geometry of infeasible perturbations of a conic linear system. SIAM J. Optim. (2000) 10:534–550CrossrefGoogle Scholar
  • Peña J. A characterization of the distance to infeasibility under block-structured perturbations. Linear Algebra Appl. (2003) 370:193–216CrossrefGoogle Scholar
  • Peña J., Renegar J. Computing approximate solutions for conic systems of constraints. Math. Programming (2000) 87:351–383CrossrefGoogle Scholar
  • Renegar J. Some perturbation theory for linear programming. Math. Programming (1994) 65:73–91CrossrefGoogle Scholar
  • Renegar J. Linear programming, complexity theory and elementary functional analysis. Math. Programming (1995) 70:279–351CrossrefGoogle Scholar
  • Renegar J. Condition numbers, the barrier method, and the conjugate-gradient method. SIAM J. Optim. (1996) 6:879–912CrossrefGoogle Scholar
  • Rohn J. Systems of linear interval equations. Linear Algebra Appl. (1989) 126:39–78CrossrefGoogle Scholar
  • Rump S. Ill-conditioned matrices are componentwise near to singularity. SIAM Rev. (1999) 41:102–112CrossrefGoogle Scholar
  • Stewart G. W. On scaled projections and pseudoinverses. Linear Algebra Its Appl. (1989) 112:189–193CrossrefGoogle Scholar
  • Todd M. J. A Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm. Oper. Res. (1990) 38:1006–1018LinkGoogle Scholar
  • Vavasis S. A., Ye Y. Condition numbers for polyhedra with real number data. Oper. Res. Lett. (1995) 17:209–214CrossrefGoogle Scholar
  • Vavasis S. A., Ye Y. A primal-dual interior point method whose running time depends only on the constraint matrix. Math. Programming (1996) 74:79–120CrossrefGoogle Scholar
  • Vera J. R. On the complexity of linear programming under finite precision arithmetic. Math. Programming (1998) 80:91–123CrossrefGoogle Scholar
  • Ye Y. Toward probabilistic analysis of interior-point algorithms for linear programming. Math. Oper. Res. (1994) 19:38–52LinkGoogle Scholar
  • Zhou K., Doyle J., Glover K.Robust and Optimal Control (1996) (Prentice Hall, Upper Saddle River, NJ) Google Scholar
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