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On strata of degenerate polyhedral cones I: Condition and distance to strata

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  • Cheung, Dennis
  • Cucker, Felipe
  • Pea, Javier

Abstract

Systems Ay[greater-or-equal, slanted]0 with a degenerate cone of solutions are considered ill-posed since finite-precision algorithms are not expected to find points in the cone of solutions. Consequently, common condition numbers for these systems, such as C(A) [J. Renegar. Some perturbation theory for linear programming, Mathematical Programming 65 (1994) 73-91] and [D. Cheung, F. Cucker, A new condition number for linear programming, Mathematical Programming 91 (2001) 163-174], which are based on the notion of distance to the nearest ill-posed problem, become infinite on such ill-posed instances. In this paper, we extend these two condition numbers to versions and which are always finite. Both condition numbers can be expressed in terms of a distance to a change in the geometry of the cone of solutions. The main result shows that for both of them, the distance corresponds to a notion of best conditioned solution for a canonical complementarity problem associated to the system Ay[greater-or-equal, slanted]0.

Suggested Citation

  • Cheung, Dennis & Cucker, Felipe & Pea, Javier, 2009. "On strata of degenerate polyhedral cones I: Condition and distance to strata," European Journal of Operational Research, Elsevier, vol. 198(1), pages 23-28, October.
    • Handle: RePEc:eee:ejores:v:198:y:2009:i:1:p:23-28
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    References listed on IDEAS

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    1. Dennis Cheung & Felipe Cucker & Javier Peña, 2003. "Unifying Condition Numbers for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 609-624, November.
    2. Yinyu Ye, 1994. "Toward Probabilistic Analysis of Interior-Point Algorithms for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 38-52, February.
    3. J. L. Goffin, 1980. "The Relaxation Method for Solving Systems of Linear Inequalities," Mathematics of Operations Research, INFORMS, vol. 5(3), pages 388-414, August.
    4. Ordónez, Fernando & Freund, Robert M., 2003. "Computational Experience and the Explanatory Value of Condition Numbers for Linear Optimization," Working papers 4337-02, Massachusetts Institute of Technology (MIT), Sloan School of Management.
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