标题： 多人口Wardrop平衡的Hessian黎曼流 显示英文标题

标题： Hessian Riemannian Flow For Multi-Population Wardrop Equilibrium

摘要： 本文研究了具有多个入口点和多个人群的广义图上的流优化问题，每人群的成本结构各不相同。 我们通过考虑由变分不等式定义的多人群Wardrop平衡来解决此问题。 我们严格分析了Wardrop平衡的存在性和唯一性。 此外，我们引入了一种有效的数值方法以求解该问题。 特别是，我们将平衡问题重新表述为子图上的分布式优化问题，并引入了一种新颖的Hessian黎曼流方法（即黎曼流形投影的Hessian流），以高效计算解。 最后，我们通过城市交通管理中的实例展示了我们的方法的有效性，包括不同类型车辆的路径规划以及在拥堵环境中减少排放的策略。 显示英文摘要

摘要： In this paper, we address the problem of optimizing flows on generalized graphs that feature multiple entry points and multiple populations, each with varying cost structures. We tackle this problem by considering the multi-population Wardrop equilibrium, defined through variational inequalities. We rigorously analyze the existence and uniqueness of the Wardrop equilibrium. Furthermore, we introduce an efficient numerical method to find the solution. In particular, we reformulate the equilibrium problem as a distributed optimization problem over subgraphs and introduce a novel Hessian Riemannian flow method, a Riemannian-manifold-projected Hessian flow, to efficiently compute a solution. Finally, we demonstrate the effectiveness of our approach through examples in urban traffic management, including routing for diverse vehicle types and strategies for minimizing emissions in congested environments.