Optimal Transport with Constraints: From Mirror Descent to Classical Mechanics

Area of Science:

• Transportation science
• Applied mathematics
• Classical mechanics

Background:

• Optimal transport theory is a powerful tool for traffic network analysis.
• Existing methods struggle to incorporate realistic network constraints.
• Congested networks present significant challenges for optimal trajectory planning.

Purpose of the Study:

• To develop a principled and flexible approach for incorporating constraints into optimal transport problems.
• To address limitations in applying optimal transport theory to realistic transportation scenarios.
• To enhance the tractability of solving optimal trajectory problems in congested networks.

Main Methods:

• A physics-based approach utilizing the D'Alembert-Lagrange principle.
• Integration of constraints into mirror descent dynamics.
• Development of a sparse, local, and linear approximation of the feasible set.

Main Results:

• Successful flexible incorporation of realistic constraints in optimal transport.
• Demonstration of a principled method derived from classical mechanics.
• Achieved closed-form updates in many optimal transport problem instances.

Conclusions:

• The proposed physics-based method offers a flexible and principled way to handle constraints in optimal transport.
• This approach enhances the applicability of optimal transport theory to complex, real-world traffic networks.
• The method's efficiency is demonstrated through sparse, local, and linear approximations leading to closed-form solutions.