2019/03/23 by Petrache, Mircea
#26D15 #28A35 #49K30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.1903.09817
In the setting of optimal transport with N≥ 2 marginals, a necessary condition for transport plans to be optimal is that they are c-cyclically monotone. For N=2 there exist several proofs that in very general settings c-cyclical monotoncity is also sufficient for optimality, while for N≥ 3 this is only known under strong conditions on c. Here we give a counterexample which shows that c-cylclical monotonicity is in general not sufficient for optimality if N≥ 3. Comparison with the N=2 case shows how the main proof strategies valid for the case N=2 might fail for N≥ 3. We leave open the question of what is the optimal condition on c under which c-cyclical monotonicity is sufficient for optimality. The new concept of an N-flow seems to be helpful for understanding the counterexample: our construction is based on the absence of finite-support N-cycles in the set where our counterexample cost c is finite. To follow this idea we formulate a Smirnov-type decomposition for N-flows.