[Paper Review] On the Monge-Kantorovich problem with additional linear constraints

Danila Zaev|arXiv (Cornell University)|Apr 19, 2014

Geometry and complex manifolds9 references4 citations

TL;DR

This paper extends the classical Monge-Kantorovich optimal transport problem by introducing additional linear constraints on transport plans, represented as vanishing integrals over a functional subspace W. It establishes a Kantorovich-type duality, proves existence of solutions under regularity conditions, and introduces a new geometric condition—(c,W)-monotonicity—characterizing optimal plans. The framework unifies important cases like martingale and invariant measures under a single theoretical umbrella.

ABSTRACT

We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal plans need to satisfy. Our main results are quite general and include several important examples. In particular, they include Monge-Kantorovich problems in the classes of invariant measures and martingales. We formulate and prove a criterion for existence of a solution, a duality statement of the Kantorovich type, and a necessary geometric condition on a support of optimal measure, which is analogues to the usual $c$-monotonicity.

Motivation & Objective

• To generalize the Monge-Kantorovich problem by incorporating linear constraints on transport plans via a functional subspace W.
• To establish existence conditions for optimal transport plans under such constraints.
• To develop a Kantorovich-type duality statement for the constrained problem.
• To introduce and characterize a new geometric condition—(c,W)-monotonicity—on the support of optimal plans.
• To unify and generalize known results on martingale and invariant measure transport problems within a single framework.

Proposed method

• Formulates the constrained Monge-Kantorovich problem as minimizing the cost ∫cdπ over transport plans π with fixed marginals μk and ∫ωdπ = 0 for all ω ∈ W.
• Introduces the functional space CL(μ) and associated seminorm ‖·‖L to handle integrability and topology of the problem.
• Applies measure-theoretic tools and Riesz representation theorems to identify linear functionals on CL(μ) with Borel probability measures on the product space X.
• Uses the stronger ‖·‖D seminorm to ensure continuity and density arguments, proving that the extended functional P on CL(μ) corresponds to a Borel probability measure π with correct marginals.
• Derives a duality result: inf ∫cdπ = sup{∑∫fk dμk : ∑fk + ω ≤ c, ω ∈ W}, linking primal and dual formulations.
• Introduces (c,W)-monotonicity as a necessary geometric condition on the support of optimal plans, generalizing classical c-monotonicity.

Experimental results

Research questions

• RQ1 Under what conditions does a solution exist for the Monge-Kantorovich problem with additional linear constraints?
• RQ2 How can a Kantorovich-type duality be formulated and proven in the presence of such constraints?
• RQ3 What geometric condition generalizes c-monotonicity when constraints are imposed via a subspace W?
• RQ4 How do known problems like martingale and invariant measure transport fit into this generalized framework?
• RQ5 Can the theory be applied to derive new results for specific cases, such as compact group invariance?

Key findings

• A solution to the constrained Monge-Kantorovich problem exists if and only if the set of admissible transport plans is non-empty, provided the cost function satisfies an appropriate regularity condition.
• A Kantorovich-type duality holds: the primal infimum equals the dual supremum over functions satisfying ∑fk + ω ≤ c for ω ∈ W.
• The support of any optimal plan must be (c,W)-monotone, a generalization of classical c-monotonicity that incorporates the constraint subspace W.
• The martingale condition arises as a special case of the linear constraint ∫ω dπ = 0, with ω being functions of the form f(x1,…,xk) − E[f(x1,…,xk+1)|x1,…,xk] in L1.
• For compact group invariance, the theory yields new results, including the existence of invariant optimal plans under suitable conditions.
• The framework unifies and extends previous results on invariant and martingale transport, showing that the general theory subsumes these cases.