Localized Normal Maps and the Stability of Variational Conditions

This paper introduces a localized version of the normal maps previously employed by the author and others to study solutions of variational inequalities. This localization permits us to deal with variational conditions posed over sets that may not be convex, and to draw conclusions about the existence and continuity of solutions of the variational condition by studying those of the normal map.As an application of these localized normal maps we investigate solutions of variational conditions when the set over which the condition is posed, as well as the function appearing in the condition, may vary continuously. In this situation solutions need be neither locally single-valued nor Lipschitzian in the parameters. Nonetheless, the normal-map approach permits us to draw fairly strong conclusions about the existence and continuity of solutions of perturbed problems if the unperturbed problems satisfy certain regularity conditions and the perturbations satisfy mild continuity requirements.

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