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In this paper, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We first derive an approximate evolution equation for the macroscopic…
A measure theoretical approach is presented to study the Monge-Kantorovich optimal mass transport problem. This approach together with Kantorovich duality provide an effective tool to answer a long standing question about the support of…
We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical…
The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod…
Planning problems are hard, motion planning, for example, isPSPACE-hard. Such problems are even more difficult in the presence of uncertainty. Although, Markov Decision Processes (MDPs) provide a formal framework for such problems, finding…
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex…
We propose a fast algorithm to approximate the optimal transport distance. The main idea is to add a Fisher information regularization into the dynamical setting of the problem, originated by Benamou and Brenier. The regularized problem is…
We propose a simple subsampling scheme for fast randomized approximate computation of optimal transport distances. This scheme operates on a random subset of the full data and can use any exact algorithm as a black-box back-end, including…
We study the stochastic optimization problem from a continuous-time perspective, with a focus on the Stochastic Gradient Descent with Momentum (SGDM) method. We show that the trajectory of SGDM, despite its \emph{stochastic} nature,…
We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semi-discrete optimal transport problem with storage fees, corresponding to a problem with hard capacity constraints. This is a variant of the…
In this paper, we present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy…
Trajectory optimization is an efficient approach for solving optimal control problems for complex robotic systems. It relies on two key components: first the transcription into a sparse nonlinear program, and second the corresponding solver…
We analyze the performance of the alternating direction method of multipliers (ADMM) to track, in a decentralized manner, a solution of a stochastic sequence of optimization problems parametrized by a discrete time Markov process. The main…
We shall present a measure theoretical approach for which together with the Kantorovich duality provide an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function…
The discretization of optimal transport problems often leads to large linear programs with sparse solutions. We derive error estimates for the approximation of the problem using convex combinations of Dirac measures and devise an active-set…
Trajectory optimization is a fundamental problem in robotics. While optimization of continuous control trajectories is well developed, many applications require both discrete and continuous, i.e., hybrid, controls. Finding an optimal…
The task of sampling from a probability density can be approached as transporting a tractable density function to the target, known as dynamical measure transport. In this work, we tackle it through a principled unified framework using…
Convergence analysis of accelerated first-order methods for convex optimization problems are presented from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient flow, has…
This paper explores expressing the relative state in the close-proximity satellite relative motion problem in terms of fundamental solution constants. The nominal uncontrolled relative state can be expressed in terms of a weighted sum of…
Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this…