Abstract

Let us say that a convex function f ⁣:C→[−∞,∞]f\colon C\to[-\infty,\infty] on a convex set C⊆RC\subseteq\mathbb{R} is infimum-stable if, for any sequence (fn)(f_n) of convex functions fn ⁣:C→[−∞,∞]f_n\colon C\to[-\infty,\infty] converging to ff pointwise, one has inf⁡Cfn→inf⁡Cf.\inf\limits_C f_n\to\inf\limits_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 (fν)(f_\nu) in place of sequences (fn)(f_n). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.

Author information

Contact details are reproduced from the original publication and may be historical.

Iosif Pinelis

Dept. of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A.

ipinelis@mtu.edu

Keywords

  • Convex functions
  • minimization
  • stability
  • convergence
  • Legendre-Fenchel transform

Mathematics Subject Classification

26A51, 90C25; 49J45, 49K05, 49K30

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Suggested citation

I. Pinelis. “A Necessary and Sufficient Condition on the Stability of the Infimum of Convex Functions.” Journal of Convex Analysis 26 (2019), No. 1, 77–87. https://doi.org/10.68381/jca26005

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.