Entropic Optimal Transport Problem with Convex Functional Cost

Abstract

We study an entropic optimal transport problem in which the transport plan is penalized by a nonlinear convex functional acting on the coupling. We establish existence, uniqueness, and uniform a priori bounds for minimizers, and we show that each minimizer satisfies a fixed-point first-order optimality system associated with an exponentially tilted reference measure. Building on this variational structure, we introduce the Sinkhorn-Frank-Wolfe (SFW) flow, prove its global well-posedness, and derive an energy-dissipation inequality yielding exponential convergence toward the unique optimal transport plan. As an application, we implement the SFW algorithm to solve an optimal routing problem for unmanned aerial vehicles with congestion aversion.