Anthony Quinn, Sarah Boufelja Y., Martin Corless, Robert Shorten, Fully probabilistic design for optimal transport, Vol. 2025 (2025), Article ID 51, pp. 1-17

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DOI: 10.23952/cot.2025.51

Received September 17, 2024; Accepted December 1, 2024; Published online December 24, 2025

Abstract. The relationship between entropy-regularized Kantorovich optimal transport (OT) and fully probabilistic design (FPD) of probability models is derived. In FPD, the (unattainable) zero-cost plan (i.e. probability measure) -called the ideal, $\pi_{I}$ -is projected (in a minimum KLD sense) into the set of feasible plans constrained by fixed marginals, $\mu$ and $\nu$ . We show that $\pi_{I}$ has a Gibbs structure. The regularizing measure, $\phi$ , acts as its base measure, and the cost metric, $c$ , acts as its energy term. Important insights and design opportunities flow from this FPD-OT setting: (i) the fixed objects in regularized OT are classified either as constraints on the actual transport plan ( $\mu$ , $\nu$ ) or else as constraints on the ideal plan ( $\phi$ , $c$ and regularization constant, $\epsilon$ ); and (ii) the modulation of $\phi$ by $c$ and $\epsilon$ , in the ideal plan, $\pi_{I}$ , allows a $c$ -dependent $\phi$ to be designed, favouring plans that meet detailed cost-dependent constraints. Extensive examples are presented, illustrating both of these insights. In particular, we show how the FPD-OT setting of discrete regularized OT allows high-cost transport paths to be quenched.

How to Cite this Article:
A. Quinn, Sarah Boufelja Y., M. Corless, Robert Shorten, Fully probabilistic design for optimal transport, Commun. Optim. Theory 2025 (2025) 51.