Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
The Monge-Kantorovich problem dates back to 1781, when Gaspard Monge set about determining the most efficient way to move a pile of dirt into a hole. It has become a famous optimization problem, with applications appearing in economics, meteorology, astrophysics, probability, and image analysis. In particular, in the case of images, the solution is a mapping that contains infinite-dimensional information about the relationship between a given pair of images. The problem is challenging to solve, as it is highly nonlinear with a complicated nonlinear constraint. We show that it can be transformed into a simpler, unconstrained problem, and show how to solve it using a simple, gradient descent algorithm.
