Calculus On Wasserstein Space And Its Applications On Distributionally Robust Optimization

Distributionally robust optimization attracts huge attention due to its wide application in the real world.In this thesis,we consider a special form of the distributionally robust optimization,which is related to the Wasserstein space.By using the calculation of the gradient flow and the geodesic,we get the properties of the function over the Wasserstein space.Hence that we introduce two different kinds of algorithms to estimate the gradient flow.The first one is the classical algorithm induced by the Euler's forward approximation.And the other is the Euler's backward approximation.Due to the structure of the Wasserstein space,these two algorithms are quite different.Euler's forward approximation is much simpler compared with the backward approximation.Furthermore,it's more convenient when we face the robust optimization problem.However,it's rather difficult to do the numerical calculation in reality when it turns to the general problem.Euler's backward approximation is dirty than the forward one.Since it contains the estimation of the continuity ODE.But backward approximation performs well in the general case.In this thesis,we focus on the Euler's forward approximation.Finally,we consider the applications on the real world.