New Structured Matrix Methods for Real and Complex Polynomial Root-finding

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular numerical approximation of the real roots of a polynomial. Our analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page

Trade Policy and Economic Integration in a Cournot Duopoly Model

This paper investigates the policy and welfare implications of forming an economic region in the context of a Cournot duopoly model. Some theoretical results are obtained. First, the economic region lowers the external tariff (against non-partner countries) less than its pre-integration level when a sufficiently large subsidy on the imports from the partner is carried out. Second, economic integration reduces the non-partner country’s welfare. Third, although the region still gains from integration even under some partial trade liberalisation regimes, complete trade liberalisation within the region leads to higher regional welfare. Finally, trade liberalisation within the region improves the welfare of the world as a whole.Trade Policy, Economic Integration

Towards Optimally Decentralized Multi-Robot Collision Avoidance via Deep Reinforcement Learning

Developing a safe and efficient collision avoidance policy for multiple robots is challenging in the decentralized scenarios where each robot generate its paths without observing other robots' states and intents. While other distributed multi-robot collision avoidance systems exist, they often require extracting agent-level features to plan a local collision-free action, which can be computationally prohibitive and not robust. More importantly, in practice the performance of these methods are much lower than their centralized counterparts. We present a decentralized sensor-level collision avoidance policy for multi-robot systems, which directly maps raw sensor measurements to an agent's steering commands in terms of movement velocity. As a first step toward reducing the performance gap between decentralized and centralized methods, we present a multi-scenario multi-stage training framework to find an optimal policy which is trained over a large number of robots on rich, complex environments simultaneously using a policy gradient based reinforcement learning algorithm. We validate the learned sensor-level collision avoidance policy in a variety of simulated scenarios with thorough performance evaluations and show that the final learned policy is able to find time efficient, collision-free paths for a large-scale robot system. We also demonstrate that the learned policy can be well generalized to new scenarios that do not appear in the entire training period, including navigating a heterogeneous group of robots and a large-scale scenario with 100 robots. Videos are available at https://sites.google.com/view/drlmac