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