Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Convex Formulation of Game-theoretic Hierarchical Routing

Dong Ho Lee, Kaitlyn Donnel, Max Z. Li, David Fridovich-Keil

TL;DR

The paper addresses hierarchical routing for multiple vehicles by coupling a discrete upper-level routing decision with a continuous, potentially noncooperative, lower-level trajectory game. It develops a convex reformulation that preserves lower-level convexity via auxiliary variables, converting the problem into a MIQP solvable by a specialized branch-and-bound algorithm, with global optimality guarantees under Slater’s condition. The contributions include (i) a mixed-integer bilevel formulation, (ii) an auxiliary-variable reformulation yielding a convex MIQP, (iii) a globally convergent branch-and-bound solver, and (iv) demonstrations on 2- and 3-vehicle formation scenarios showing how lower-level interactions influence high-level routing. The approach advances integrated routing and trajectory planning in air-traffic-like settings, with potential impact on real-time automation systems such as FMDS by enabling coordinated, dynamics-aware routing decisions.

Abstract

Hierarchical decision-making is a natural paradigm for coordinating multi-agent systems in complex environments such as air traffic management. In this paper, we present a bilevel framework for game-theoretic hierarchical routing, where a high-level router assigns discrete routes to multiple vehicles who seek to optimize potentially noncooperative objectives that depend upon the assigned routes. To address computational challenges, we propose a reformulation that preserves the convexity of each agent's feasible set. This convex reformulation enables a solution to be identified efficiently via a customized branch-and-bound algorithm. Our approach ensures global optimality while capturing strategic interactions between agents at the lower level. We demonstrate the solution concept of our framework in two-vehicle and three-vehicle routing scenarios.

A Convex Formulation of Game-theoretic Hierarchical Routing

TL;DR

The paper addresses hierarchical routing for multiple vehicles by coupling a discrete upper-level routing decision with a continuous, potentially noncooperative, lower-level trajectory game. It develops a convex reformulation that preserves lower-level convexity via auxiliary variables, converting the problem into a MIQP solvable by a specialized branch-and-bound algorithm, with global optimality guarantees under Slater’s condition. The contributions include (i) a mixed-integer bilevel formulation, (ii) an auxiliary-variable reformulation yielding a convex MIQP, (iii) a globally convergent branch-and-bound solver, and (iv) demonstrations on 2- and 3-vehicle formation scenarios showing how lower-level interactions influence high-level routing. The approach advances integrated routing and trajectory planning in air-traffic-like settings, with potential impact on real-time automation systems such as FMDS by enabling coordinated, dynamics-aware routing decisions.

Abstract

Hierarchical decision-making is a natural paradigm for coordinating multi-agent systems in complex environments such as air traffic management. In this paper, we present a bilevel framework for game-theoretic hierarchical routing, where a high-level router assigns discrete routes to multiple vehicles who seek to optimize potentially noncooperative objectives that depend upon the assigned routes. To address computational challenges, we propose a reformulation that preserves the convexity of each agent's feasible set. This convex reformulation enables a solution to be identified efficiently via a customized branch-and-bound algorithm. Our approach ensures global optimality while capturing strategic interactions between agents at the lower level. We demonstrate the solution concept of our framework in two-vehicle and three-vehicle routing scenarios.

Paper Structure

This paper contains 16 sections, 1 theorem, 10 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Preliminaries
  5. Proposed Bilevel Framework
  6. Upper-level Routing Problem
  7. Lower-level Trajectory Game
  8. Structure of Cost Functions
  9. Methodology
  10. kkt Reformulation
  11. Introducing Auxillary Variables to Preserve Convexity
  12. Branch-and-bound algorithm
  13. Experimental Results
  14. Experiment Setup
  15. Detailed Analysis of Results
  16. ...and 1 more sections

Key Result

Proposition 1

Under the following conditions, branch-and-bound finds a global optimal solution to eqn:IV-method-bilevel-auxillary-reform:

Figures (2)

  • Figure 1: Comparison of optimal vehicle routes and trajectories in a two-vehicle formation flight routing game. Dashed straight lines indicate each vehicle’s visited nodes.
  • Figure 2: Comparison of optimal vehicle routes and trajectories in a three-vehicle formation flight routing game. Dashed straight lines indicate each vehicle’s visited nodes.

Theorems & Definitions (4)

  • Definition 4.1: Slater's condition for the lower level
  • Proposition 1
  • proof
  • Remark 1