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

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Abstract

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.

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Authors and Affiliations

  1. Operations Research Center, Miguel Hernández University of Elche, Elche, Alicante, Spain

    M. J. Cánovas (Associate Professor), J. Parra (Associate Professor) & F. J. Toledo (Associate Professor)

  2. Department of Statistics and Operations Research, University of Alicante, Alicante, Spain

    M. A. López (Professor)

Authors
  1. M. J. Cánovas
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  2. M. A. López
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  3. J. Parra
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  4. F. J. Toledo
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Communicated by J.P. Crouzeix

This research has been partially supported by grants BFM2002–04114-C02 (01–02) from MEC (Spain) and FEDER (EU) and by grants GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain).

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Cánovas, M.J., López, M.A., Parra, J. et al. Distance to Ill-Posedness in Linear Optimization via the Fenchel-Legendre Conjugate. J Optim Theory Appl 130, 173–183 (2006). https://doi.org/10.1007/s10957-006-9097-5

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  • DOI: https://doi.org/10.1007/s10957-006-9097-5

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