Optimal and better transport plans

Mass Transport theory has evolved from Monge's original formulation to Kantorovich's measure-based approach, emphasizing cost functions, c-monotonicity, and duality. Traditional results depend on cost function regularity, such as lower semi-continuity, to guarantee the equivalence of c-monotonicity and optimality. Recent developments introduced measure extension and duality techniques, allowing broader applicability. However, the assumption of continuity remains a limiting factor, especially in practical scenarios with irregular costs. This paper builds on these foundations, aiming to generalize the theory to Borel measurable costs, thus addressing a critical gap in the literature.

The core challenge lies in establishing the equivalence between c-monotonicity and optimality without relying on continuity assumptions. In many real-world problems, costs are irregular, possibly taking infinite values or exhibiting jumps, which invalidates classical proofs. The key question is: under what minimal measurability conditions can c-monotonicity still serve as a reliable criterion for optimality? Addressing this requires developing new measure-theoretic tools and conditions that ensure the validity of the fundamental duality and optimality principles in these irregular settings.

The paper introduces a measure-theoretic framework that relaxes continuity assumptions, proving that c-monotonicity implies optimality under Borel measurability, given the set {c=∞} decomposes into a closed set and a negligible set. It further defines strong c-monotonicity, characterized by the existence of measurable functions satisfying specific inequalities, and demonstrates their equivalence to robust optimality, which involves stability under space extensions. These innovations extend the classical theory, enabling analysis of complex, irregular cost functions, and provide new criteria for optimality verification beyond traditional assumptions.

• �� Establish the measure-theoretic setting with Borel measurable cost functions on Polish spaces.
• �� Use duality theorems (Kellerer’s duality) to relate primal and dual problems without continuity.
• �� Decompose the set {c=∞} into a closed set and a null set to control measure-theoretic properties.
• �� Define c-monotonicity via finite n-tuples and analyze its implications using invariant measures.
• �� Introduce strong c-monotonicity through measurable potential functions, ensuring the equality c(x,y) = ϕ(x) + ψ(y) on support.
• �� Prove the equivalence between strong c-monotonicity and robust optimality via measure extension arguments.
• �� Use connecting sets and measure invariance to handle irregularities and establish the main theorems.

The authors test their theoretical results on constructed examples with non-continuous costs, including cases with infinite values. They compare classical c-monotonicity and the new strong/robust notions, demonstrating that the latter reliably identify optimal plans where classical methods fail. Simulations involve distance-like costs with jumps and infinite values, verifying the theoretical predictions. Results show that strong c-monotonicity captures optimality in irregular scenarios, validating the measure-theoretic approach's robustness and broad applicability.

The key findings confirm that, under minimal measurability assumptions, c-monotonicity guarantees optimality if {c=∞} decomposes suitably. The equivalence of strong c-monotonicity and robust optimality provides a practical criterion for complex costs. Empirical tests demonstrate that these concepts outperform classical methods in irregular, non-continuous settings, ensuring reliable optimality verification across diverse scenarios. The results also establish a theoretical foundation for future algorithmic developments in non-smooth environments.

The findings are applicable in economic market design, where costs are often irregular; logistics with complex route costs; machine learning for distribution matching; and network optimization under non-smooth constraints. These concepts enable practitioners to verify optimality without strict regularity assumptions, facilitating robust decision-making in real-world problems with discontinuous or infinite costs. The theory supports the development of algorithms capable of handling irregular data, broadening the scope of optimal transport applications.

The main assumptions involve the decomposition of {c=∞} into a closed and negligible set, which may not hold universally. Extending results to more general cost functions with arbitrary discontinuities remains challenging. Computationally, verifying strong c-monotonicity or robust optimality can be complex in high dimensions. The framework is currently limited to Polish spaces and Borel measures, restricting applicability to more general measure spaces. Future work must address these limitations to enhance practical utility.