On Condition Number Theorems in Mathematical Programming

Tullio Zolezzi ()
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Tullio Zolezzi: University of Genova

Journal of Optimization Theory and Applications, 2017, vol. 175, issue 3, No 1, 597-623

Abstract: Abstract A condition number of nonconvex mathematical programming problems is defined as a measure of the sensitivity of their global optimal solutions under canonical perturbations. A (pseudo-)distance among problems is defined via the corresponding augmented Kojima functions. A characterization of well-conditioning is obtained. In the nonconvex case, we prove that the distance from ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number. Moreover, a lower bound of the distance from a special class of ill-conditioned problems is obtained in terms of the condition number. The proof is based on a new theorem about the permanence of the Lipschitz character of set-valued inverse mappings. A uniform version of the condition number theorem is proved for classes of convex problems defined through bounds of some constants available from problem’s data.

Keywords: Condition number; Condition number theorem; Mathematical programming problems with canonical perturbations; 90C25; 90C30; 90C31 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-017-1191-3

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