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A Rockafellar-type theorem for non-traditional costs

Shiri Artstein-Avidan, Shay Sadovsky, Katarzyna Wyczesany

TL;DR

The paper addresses the existence of a $c$-potential in optimal transport when costs may take infinite values, introducing the key concept of $c$-path-boundedness. It reduces the problem to solvability of a (potentially infinite) system of linear inequalities and distinguishes between countable and uncountable index sets, with an additional no-infinite-black-hole condition in the latter. The main result shows that a $c$-path-bounded set $G$ (countable, or without infinite black holes) is contained in the $c$-subgradient of a $c$-class function, thus extending Rockafellar-type theory to non-traditional costs; in the finite-valued case this recovers the classical Rockafellar-Rochet-Rüschendorf theorem. The work also provides an elementary proof pathway for the classical results and discusses special cases, including polar costs, clarifying when $c$-cyclic monotonicity suffices to guarantee a $c$-potential.

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 that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call $c$-path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be $c$-path-bounded, and this condition 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.

A Rockafellar-type theorem for non-traditional costs

TL;DR

The paper addresses the existence of a -potential in optimal transport when costs may take infinite values, introducing the key concept of -path-boundedness. It reduces the problem to solvability of a (potentially infinite) system of linear inequalities and distinguishes between countable and uncountable index sets, with an additional no-infinite-black-hole condition in the latter. The main result shows that a -path-bounded set (countable, or without infinite black holes) is contained in the -subgradient of a -class function, thus extending Rockafellar-type theory to non-traditional costs; in the finite-valued case this recovers the classical Rockafellar-Rochet-Rüschendorf theorem. The work also provides an elementary proof pathway for the classical results and discusses special cases, including polar costs, clarifying when -cyclic monotonicity suffices to guarantee a -potential.

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 that relies on proving solvability of a special (possibly infinite) family of linear inequalities. When the index set of this family is countable, we give a necessary and sufficient condition on the coefficients that assures the existence of a solution, and which, in the setting of transport theory, we call -path-boundedness. In the case of an uncountable index set, one needs an additional assumption for solvability. We propose a sufficient condition in this case. We note that any set admitting a potential must be -path-bounded, and this condition replaces -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.

Paper Structure

This paper contains 21 sections, 12 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Costs and potentials
  3. Mass transport
  4. The $c$-class and basic functions
  5. The $c$-subgradient
  6. $c$-cyclic monotonicity
  7. Example: The quadratic cost
  8. The Rockafellar-Rochet-Rüschendorf Theorem
  9. Known results
  10. An example of a $c$-cyclically monotone set with no $c$-potential
  11. A Rockafellar-type result for non-traditional costs
  12. Reformulation of the problem
  13. Solutions for families of linear inequalities
  14. The countable case
  15. Graph theoretical component of the proof
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X,\,Y$ be two arbitrary sets and $c:X\times Y\to (-\infty, \infty]$ an arbitrary cost function. Assume that $G\subset X\times Y$ is a $c$-path-bounded subset that is countable, or if it is uncountable, it does not have an infinite black hole. Then there exists a $c$-class function $\varphi:X\to

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Definition 1.1
  • Theorem 3.1
  • proof
  • ...and 19 more