Abstract

Suppose f(x,y)+κ2∥x∥2−σ2∥y∥2f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2 is convex where κ≥0,σ>0\kappa\ge 0, \sigma>0, and the argmin function γ(x)={γ:inf⁡yf(x,y)=f(x,γ)}\gamma(x) = \{ \gamma: \inf_y f(x,y) = f(x,\gamma)\} exists and is single valued. We will prove γ\gamma is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

Author information

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

Julius Ross

Dept. of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, U.S.A.

julius@math.uic.edu

David Witt Nyström

Dept. of Mathematical Sciences, University of Gothenburg, 41296 Göteborg, Sweden

david.witt.nystrom@gu.se

Keywords

  • Argmin function
  • differentiability
  • minimum principle
  • semiconcave subsolutions

Mathematics Subject Classification

28B20, 58C06

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Suggested citation

J. Ross, D. Witt Nyström. “Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions.” Journal of Convex Analysis 27 (2020), No. 3, 811–832. https://doi.org/10.68381/jca27041

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.