Mathematics

The nested distance builds on the Wasserstein distance to quantify the difference of stochastic processes, including also the information modelled by filtrations. The Sinkhorn divergence is a relaxation of the Wasserstein distance, which can be computed considerably faster. For this reason we employ the Sinkhorn divergence and take advantage of the related (fixed point) iteration algorithm. Furthermore, we investigate the transition of the entropy throughout the stages of the stochastic process and provide an entropy-regularized nested distance formulation, including a characterization of its dual. Numerical experiments affirm the computational advantage and supremacy.

Guaranteeing the quality of transit service is of great importance to promote the attractiveness of buses and alleviate urban traffic issues such as congestion and pollution. Emerging technologies of automated driving and V2X communication have the potential to enable the accurate control of vehicles and the efficient organization of traffic to enhance both the schedule adherence of buses and the overall network mobility. This study proposes an innovative network-level control scheme for heterogeneous automated traffic composed of buses and private cars under a full connected and automated environment. Inheriting the idea of network-level rhythmic control proposed by Lin et al. (2020), an augmented rhythmic control scheme for heterogeneous traffic, i.e., RC-H, is established to organize the mixed traffic in a rhythmic manner. Realized virtual platoons are designed for accommodating vehicles to pass through the network, including dedicated virtual platoons for buses to provide exclusive right-of-ways (ROWs) on their trips and regular virtual platoons for private cars along with an optimal assignment plan to minimize the total travel cost. A mixed-integer linear program (MILP) is formulated to optimize the RC-H scheme and a bilevel heuristic solution method is designed to relieve the computational burden of MILP. Numerical examples and simulation experiments are conducted to evaluate the performance of the RC-H scheme under different scenarios. The results show that the bus operation can be guaranteed and the travel delay can be minimized under various demand levels with transit priority. Moreover, compared with traffic signal control strategies, the RC-H scheme has significant advantages in handling massive traffic demand, in terms of both vehicle delay and network throughput.

This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and derivative pricing in financial engineering. We survey recent results in the literature, present new developments, notably in the fully nonlinear case, and compare the different schemes illustrated by numerical tests on various financial applications. We conclude by highlighting some future research directions.

We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal blocks of the Hessian of the tensor completion cost function, thus has a preconditioning effect on these algorithms. We prove that the proposed Riemannian gradient descent algorithm globally converges to a stationary point of the tensor completion problem, with convergence rate estimates using the ? ojasiewicz property. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms. Moreover, the proposed algorithms display a greater tolerance for overestimated rank parameters in terms of the tensor recovery performance, thus enable a flexible choice of the rank parameter.

When solving hard multicommodity network flow problems using an LP-based approach, the number of commodities is a driving factor in the speed at which the LP can be solved, as it is linear in the number of constraints and variables. The conventional approach to improve the solve time of the LP relaxation of a Mixed Integer Programming (MIP) model that encodes such an instance is to aggregate all commodities that have the same origin or the same destination. However, the bound of the resulting LP relaxation can significantly worsen, which tempers the efficiency of aggregating techniques. In this paper, we introduce the concept of partial aggregation of commodities that aggregates commodities over a subset of the network instead of the conventional aggregation over the entire underlying network. This offers a high level of control on the trade-off between size of the aggregated MIP model and quality of its LP bound. We apply the concept of partial aggregation to two different MIP models for the multicommodity network design problem. Our computational study on benchmark instances confirms that the trade-off between solve time and LP bound can be controlled by the level of aggregation, and that choosing a good trade-off can allow us to solve the original large-scale problems faster than without aggregation or with full aggregation.

A set of quadratic forms is simultaneously diagonalizable via congruence (SDC) if there exists a basis under which each of the quadratic forms is diagonal. This property appears naturally when analyzing quadratically constrained quadratic programs (QCQPs) and has important implications in this context. This paper extends the reach of the SDC property by studying two new related but weaker notions of simultaneous diagonalizability. Specifically, we say that a set of quadratic forms is almost SDC (ASDC) if it is the limit of SDC sets and d-restricted SDC (d-RSDC) if it is the restriction of an SDC set in up to d-many additional dimensions. Our main contributions are a complete characterization of the ASDC pairs and the nonsingular ASDC triples, as well as a sufficient condition for the 1-RSDC property for pairs of quadratic forms. Surprisingly, we show that every singular pair is ASDC and that almost every pair is 1-RSDC. We accompany our theoretical results with preliminary numerical experiments applying the RSDC property to QCQPs with a single quadratic constraint.

We propose a distributed cubic regularization of the Newton method for solving (constrained) empirical risk minimization problems over a network of agents, modeled as undirected graph. The algorithm employs an inexact, preconditioned Newton step at each agent's side: the gradient of the centralized loss is iteratively estimated via a gradient-tracking consensus mechanism and the Hessian is subsampled over the local data sets. No Hessian matrices are thus exchanged over the network. We derive global complexity bounds for convex and strongly convex losses. Our analysis reveals an interesting interplay between sample and iteration/communication complexity: statistically accurate solutions are achievable in roughly the same number of iterations of the centralized cubic Newton method, with a communication cost per iteration of the order of O ? (1/ 1?��?????????????) , where ? characterizes the connectivity of the network. This demonstrates a significant communication saving with respect to that of existing, statistically oblivious, distributed Newton-based methods over networks.

We are concerned with the tensor equations whose coefficient tensor is an M-tensor. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend the method to solve the equation with a nonnegative constant term and establish its convergence. At last, we do numerical experiments to test the proposed methods. The results show that the proposed method is quite efficient.

In recent years there has been significant effort to adapt the key tools and ideas in convex optimization to the Riemannian setting. One key challenge has remained: Is there a Nesterov-like accelerated gradient method for geodesically convex functions on a Riemannian manifold? Recent work has given partial answers and the hope was that this ought to be possible. Here we dash these hopes. We prove that in a noisy setting, there is no analogue of accelerated gradient descent for geodesically convex functions on the hyperbolic plane. Our results apply even when the noise is exponentially small. The key intuition behind our proof is short and simple: In negatively curved spaces, the volume of a ball grows so fast that information about the past gradients is not useful in the future.

We establish a generalization of Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton-Jacobi-Bellman equation associated to an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton's optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales.