Optimal transport problems regularized by generic convex functions: a geometric and algorithmic approach

In order to circumvent the difficulties in solving numerically discrete optimal transport problem, which one minimizes linear target function $$P\mapsto \left\langle C,P\right\rangle {:}{=}\sum _{i,j}C_{ij}P_{ij}$$ , Cuturi introduced a variant of problem is altered by convex $$\varPhi (P)=\left\langle -\lambda {{\mathcal {H}}}(P)$$ where $${{\mathcal {H}}}$$ Shannon entropy and $$\lambda $$ positive constant. We herein generalize their formulation form +\lambda f(P)$$ f generic strictly smooth function. also propose an iterative method for finding numerical solution, clarify that proposed particularly efficient when $$f(P)=\frac{1}{2}\Vert P\Vert ^2$$ .