A Rockafellar-type theorem for non-traditional costs

S. Artstein-Avidan, S. Sadovsky, K. Wyczesany

Research output: Contribution to journalArticlepeer-review

Abstract

In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is costs that assume infinite values. We establish a new method which relies on proving solvability of a special (possibly infinite) family of linear inequalities. We give a necessary and sufficient condition on the coefficients that assure the existence of a solution, and which in the setting of transport theory we call c-path-boundedness. This condition fully characterizes sets that admit a potential and replaces c-cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and Rüschendorf.

Original languageEnglish
Article number108157
JournalAdvances in Mathematics
Volume395
DOIs
StatePublished - 22 Jan 2022

Keywords

  • Existence of a potential
  • Optimal transport
  • Rockafellar-type theorem
  • c-cyclic monotonicity
  • c-path-boundedness

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