Journal of Optimization Theory and Applications

, Volume 130, Issue 2, pp 173–183 | Cite as

Distance to Ill-Posedness in Linear Optimization via the Fenchel-Legendre Conjugate

  • M. J. Cánovas
  • M. A. López
  • J. Parra
  • F. J. Toledo


We consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space and with a fixed index set, endowed with the topology of the uniform convergence of the coefficient vectors. A system is ill-posed with respect to the consistency when arbitrarily small perturbations yield both consistent and inconsistent systems. In this paper, we establish a formula for measuring the distance from the nominal system to the set of ill-posed systems. To this aim, we use the Fenchel-Legendre conjugation theory and prove a refinement of the formula in Ref. 1 for the distance from any point to the boundary of a convex set.


Fenchel-Legendre conjugate stability well-posedness linear inequality systems distance to ill-posedness 


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Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • M. A. López
    • 2
  • J. Parra
    • 1
  • F. J. Toledo
    • 1
  1. 1.Operations Research CenterMiguel Hernández University of Elche, ElcheAlicanteSpain
  2. 2.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain

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