Optimal solutions of multivariate coupling problems
Some necessary and some sufficient conditions are established for the explicit construction and characterization of optimal solutions of multivariate transportation (coupling) problems. The proofs are based on ideas from duality theory and nonconvex optimization theory. Applications are given to multivariate optimal coupling problems w.r.t. minimal $l_p$-type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest even in the one-dimensional case. For the first time an explicit criterion is given for the construction of optimal multivariate couplings for the Kantorovich metric $l_1$.