Abstract
A finite-dimensional mathematical programming problem with convex data and inequality constraints is considered. A suitable definition of condition number is obtained via canonical perturbations of the given problem, assuming uniqueness of the optimal solutions. The distance among mathematical programming problems is defined as the Lipschitz constant of the difference of the corresponding Kojima functions. It is shown that the distance to ill-conditioning is bounded above and below by suitable multiples of the reciprocal of the condition number, thereby generalizing the classical Eckart–Young theorem. A partial extension to the infinite-dimensional setting is also obtained.
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Acknowledgments
We thank a referee for helpful critical comments improving the presentation. Work partially supported by Università di Genova—progetti di ricerca di Ateneo.
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Zolezzi, T. A condition number theorem in convex programming. Math. Program. 149, 195–207 (2015). https://doi.org/10.1007/s10107-014-0745-5
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DOI: https://doi.org/10.1007/s10107-014-0745-5