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

Author

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  • M. J. Cánovas

    (Miguel Hernández University of Elche, Elche)

  • M. A. López

    (University of Alicante)

  • J. Parra

    (Miguel Hernández University of Elche, Elche)

  • F. J. Toledo

    (Miguel Hernández University of Elche, Elche)

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.

Suggested Citation

  • M. J. Cánovas & M. A. López & J. Parra & F. J. Toledo, 2006. "Distance to Ill-Posedness in Linear Optimization via the Fenchel-Legendre Conjugate," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 173-183, August.
    • Handle: RePEc:spr:joptap:v:130:y:2006:i:2:d:10.1007_s10957-006-9097-5
    DOI: 10.1007/s10957-006-9097-5
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    References listed on IDEAS

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    1. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
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    1. Goberna, M.A. & Jeyakumar, V. & Li, G. & Vicente-Pérez, J., 2022. "The radius of robust feasibility of uncertain mathematical programs: A Survey and recent developments," European Journal of Operational Research, Elsevier, vol. 296(3), pages 749-763.

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