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arXiv · 1307.3806

A necessary and sufficient condition on the stability of the infimum of convex functions

Iosif Pinelis
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Abstract

Let us say that a convex function f\colon C\to[-\infty,\infty] on a convex set C\subseteq\R is infimum-stable if, for any sequence (f_n) of convex functions f_n\colon C\to[-\infty,\infty] converging to f pointwise, one has \inf_C f_n\to\inf_C f. A simple necessary and sufficient condition for a convex function to be infimum-stable is given. The same condition remains necessary and sufficient if one uses Moore--Smith nets (f_ν) in place of sequences (f_n). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.

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Iosif Pinelis. 2013-08-01. A necessary and sufficient condition on the stability of the infimum of convex functions. https://arxiv.org/abs/1307.3806

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