Sinkhorn-Knopp balancing with generalised martingale-type constraints

Abstract: We consider the problem of optimally distributing resources from a set of suppliers to a set of consumers in the presence of general transportation constraints, and including heterogeneous flexibility in the marginal constraints. Such problems frequently arise in a variety of practical settings; for example, in the context of sharing economy applications, where one is not only interested in the transportation plan, but also its realisation, and in other problems that involve the study of martingales. Our principal contribution in this paper is to consider a generalisation of the classic entropically regularised Optimal Transport formulation in which such problems can be solved with a Sinkhorn algorithm. In particular, we present provably convergent Sinkhorn-like algorithms for solving this class of problems, and provide examples to both illustrate the utility of our approach as well as its efficacy.

• The paper develops convergent Sinkhorn-like algorithms that combine hard zero entries, KL-regularised reference plans, general linear constraints, and selectively enforced marginal conditions through cyclic Bregman projections.
• The framework provides a multiplicative scaling characterisation when an interior feasible solution exists, while identifying boundary cases where admissible transport plans must contain structurally zero entries.
• The numerical studies show that a ridesharing fairness constraint eliminates a 22.5% pay gap with only a 3.4% transport-cost increase, while an energy-market model allocates flexibility according to its cost under hard infrastructure restrictions.

Problem setting and motivation

This paper addresses entropically regularised optimal transport (OT) problems in which the transport plan is subject to heterogeneous marginal constraints, hard zero entries, and arbitrary linear constraints on the coupling. The motivating applications are resource-allocation problems in sharing-economy settings, where suppliers and consumers have differing contractual flexibility and where the realisation of the plan itself induces constraints on the coupling — for example, equalising earnings opportunity across driver subgroups in ride-hailing.

The classical discrete OT problem is a linear program that scales poorly; the standard remedy is entropic regularisation solved via the Sinkhorn-Knopp (SK) algorithm [Cut1]. Prior extensions to constrained OT — multi-marginal formulations, martingale couplings, sparsity or zero-transport constraints — typically either lack convergence guarantees or require strict enforcement of all marginal constraints, making them unsuitable for unbalanced problems in which marginals have unequal mass or not all mass need be transported. The paper's stated contribution is a provably convergent family of Sinkhorn-like algorithms that simultaneously supports: (i) hard zero constraints on plan entries, (ii) an ideal reference plan for KL regularisation, (iii) general linear constraints of martingale type, and (iv) selective compliance with row and column marginal constraints.

Formulation

The problem is posed over $m$ source agents with capacities $\tilde{u}_i$ and $n$ target agents with desired masses $\tilde{v}_j$. The admissible set $\mathcal{T}$ enforces nonnegativity on an index set $\mathcal{Z}$ and $t_{ij}=0$ off $\mathcal{Z}$, assuming no row or column of $\mathcal{Z}$ is empty. The cost combines linear transport cost with a KL divergence from an ideal plan $\tilde{T}$, which after completion of squares reduces to minimising $\tilde{u}_i$0 against the Gibbs kernel $\tilde{u}_i$1 on $\tilde{u}_i$2.

The generalised problem augments this objective with soft penalty terms $\tilde{u}_i$3 for $\tilde{u}_i$4, while enforcing hard linear constraints $\tilde{u}_i$5 for $\tilde{u}_i$6. This formulation subsumes capacity constraints on individual plan entries, total-mass constraints, and martingale constraints: the martingale condition $\tilde{u}_i$7 per row is exactly a constraint of the required form, justifying the paper's title. A special case treats selectively relaxed marginals: rows in $\tilde{u}_i$8 and columns in $\tilde{u}_i$9 are enforced exactly, while the remainder are penalised with weights $n$0. This strictly generalises prior work in which all row constraints were enforced and none of the column constraints were [martin].

Algorithms and convergence

Four algorithms are developed. Algorithm 1 cycles through the $n$1 constraints, at each step solving a one-dimensional scaling equation for $n$2 and applying multiplicative updates $n$3; for soft constraints ($n$4), an auxiliary variable $n$5 is scaled concurrently. Its convergence follows from Bregman's cyclic projection theorem [Bregman1967]: under feasibility and the assumption that each single-constraint projection has an interior solution, the iterates converge to a minimiser of the generalised problem (Theorem 1). Algorithm 2 re-expresses the same iteration in terms of cumulative scaling parameters $n$6, yielding a direct generalisation of SK balancing in which the optimal plan takes the form $n$7. Algorithms 3 and 4 specialise these to the selectively relaxed marginal case, with closed-form exponentiated scalings $n$8 for non-enforced rows and columns.

A key structural result concerns boundary optimisers. With hard zero constraints, the minimiser may lie on the boundary of $n$9 (the paper gives a two-source example where the unique feasible plan forces $\tilde{v}_j$0). Lemma 1 establishes that if any coordinate of a KL-minimiser over a convex set vanishes, then every feasible point has that coordinate zero; Corollary 1 transfers this to the OT setting. Consequently, if some feasible plan has $\tilde{v}_j$1 for all $\tilde{v}_j$2, the optimiser is interior and admits the exact multiplicative characterisation of Lemma 3 — the fixed-point equations that Algorithms 1–4 solve. This dichotomy cleanly separates instances where the scaling ansatz is valid from degenerate cases.

Applications

Two numerical case studies illustrate the framework. In the ridesharing example, male and female driver distributions are modelled as beta distributions concentrated near and away from the city centre respectively, with squared Euclidean distance as cost. Imposing the fairness constraint $\tilde{v}_j$3, which equalises expected fare utility across gender groups, reduces the pay gap from 22.5% under unconstrained matching to effectively zero, at a global transport-cost increase of only 3.4%. Convergence is demonstrated over 200 iterations of Algorithm 2. This quantifies the trade-off between allocative efficiency and equality of opportunity, and shows the fairness constraint can be met cheaply in this synthetic setting.

In the energy-market example, 200 inflexible suppliers serve 500 consumers whose aggregate demand exceeds supply; 75% of consumers are flexible, with heterogeneity costs drawn iid from $\tilde{v}_j$4, and 700 forbidden source-target pairs simulate infrastructure constraints. Setting column penalties proportional to negotiated flexibility costs, Algorithm 4 converges to tolerance $\tilde{v}_j$5 and produces plans in which realised flexibility is inversely proportional to its cost — i.e., the algorithm automatically minimises total flexibility expenditure while respecting all hard constraints.

Limitations and open questions

Several caveats bear directly on the results. Convergence of Algorithm 1 relies on Assumption 1, namely that each single-constraint Bregman projection admits an interior solution; the paper does not characterise when this fails in practice, nor does it provide convergence rates analogous to those known for classical SK. The interior-optimiser characterisation applies only when every admissible entry can be made positive by some feasible plan; on the constraint boundary the multiplicative structure breaks down, and the algorithms' behaviour there is not analysed. Each iteration requires solving a scalar nonlinear equation for $\tilde{v}_j$6, which is well-posed (strictly monotone) only under sign conditions on the coefficients $\tilde{v}_j$7 and targets $\tilde{v}_j$8; the general signed case is not addressed. Finally, the empirical results are confined to synthetic simulations with prescribed distributions; no comparison against alternative constrained-OT solvers (e.g., [tang2024sinkhorn]) is provided, and the robustness of the reported 3.4% optimality gap under realistic demand data remains untested.

Conclusion

The paper extends entropic OT with a unified treatment of hard zero constraints, general linear (including martingale-type) constraints, and selectively relaxed marginals, accompanied by provable convergence via Bregman projection theory and a sharp characterisation of when the Sinkhorn scaling form yields the optimum. The ridesharing and energy-market experiments demonstrate that fairness and flexibility objectives can be encoded as linear constraints and penalties within a single efficiently solvable program, with modest loss in transport efficiency.