Algorithms for constrained optimal transport

The paper formulates the same objective J(T)=KL(T|K)+γ KL(v_T|ṽ) over V(ũ), proposes Algorithm 1 with updates (19)–(22), and proves convergence via iterative KL/Bregman projections on the joint variables (T,v), using Lemmas 3–4 and a Bregman projection theorem (Theorem 2). It also derives the diagonal scaling KKT form t*_{ij}=d1_i k_{ij} d2_j with the fixed-point updates for d1,d2. The candidate solution mirrors this: it sets the joint projection problem, identifies the same KKT scalings, interprets Algorithm 1 as alternating KL projections onto the same affine constraints, and invokes the same convergence principle. A minor omission in the candidate solution is not explicitly checking the gradient-range condition used in Theorem 2; the paper handles this by choosing x(0)=vec(K,γ ṽ), for which ∇f(x(0)) lies in the required range. Otherwise the arguments agree and are complete. Key alignments: problem (17)–(18), Algorithm 1 (19)–(22), the joint formulation (48)–(49), projection lemmas, KKT scaling, and convergence via Bregman projections .