Analytic solutions for the approximated 1-D Kantorovich mass transfer problems

TL;DR

This paper develops an analytical approach using canonical duality theory to approximate the global maximizer of the 1-D Monge-Kantorovich mass transfer problem through a sequence of solvable minimization problems.

Contribution

It introduces a novel method transforming the mass transfer problem into a sequence of minimization problems and proves convergence to the global maximizer.

Findings

Derived explicit analytic solutions for the minimization problems.

Proved convergence of the solutions to the global maximizer.

Validated the approach through theoretical analysis.

Abstract

This paper mainly investigates the approximation of a global maximizer of the 1-D Monge-Kantorovich mass transfer problem through the approach of nonlinear differential equations with Dirichlet boundary. Using an approximation mechanism, the primal maximization problem can be transformed into a sequence of minimization problems. By applying the canonical duality theory, one is able to derive a sequence of analytic solutions for the minimization problems. In the final analysis, the convergence of the sequence to a global maximizer of the primal Monge-Kantorovich problem will be demonstrated.