Abstract

The Fenchel-Young inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter. In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results.

Author information

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

Heinz H. Bauschke

Department of Mathematics, University of British Columbia, Kelowna, Canada

heinz.bauschke@ubc.ca

Shambhavi Singh

Department of Mathematics, University of British Columbia, Kelowna, Canada

sambha@student.ubc.ca

Xianfu Wang

Department of Mathematics, University of British Columbia, Kelowna, Canada

shawn.wang@ubc.ca

Keywords

  • Carlier's inequality
  • cyclic monotonicity
  • Fenchel conjugate
  • Fenchel-Young inequality
  • Fitzpatrick function
  • maximally monotone operator
  • proximal mapping
  • resolvent

Mathematics Subject Classification

26B25, 47H05; 26D07, 90C25

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Suggested citation

H. H. Bauschke, S. Singh, X. Wang. “On Carlier's Inequality.” Journal of Convex Analysis 30 (2023), No. 2, 499–514. https://doi.org/10.68381/jca30029

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.