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

Author： Cánovas M.   López M.   Parra J.   Toledo F.  

Publisher： Springer Publishing Company

ISSN： 0022-3239

Source： Journal of Optimization Theory and Applications, Vol.130, Iss.2, 2006-08, pp. : 173-183

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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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