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A Mixed-Integer Conic Program for the Moving-Target Traveling Salesman Problem based on a Graph of Convex Sets

Allen George Philip, Zhongqiang Ren, Sivakumar Rathinam, Howie Choset

Abstract

This paper introduces a new formulation that finds the optimum for the Moving-Target Traveling Salesman Problem (MT-TSP), which seeks to find a shortest path for an agent, that starts at a depot, visits a set of moving targets exactly once within their assigned time-windows, and returns to the depot. The formulation relies on the key idea that when the targets move along lines, their trajectories become convex sets within the space-time coordinate system. The problem then reduces to finding the shortest path within a graph of convex sets, subject to some speed constraints. We compare our formulation with the current state-of-the-art Mixed Integer Conic Program (MICP) solver for the MT-TSP. The experimental results show that our formulation outperforms the MICP for instances with up to 20 targets, with up to two orders of magnitude reduction in runtime, and up to a 60\% tighter optimality gap. We also show that the solution cost from the convex relaxation of our formulation provides significantly tighter lower bounds for the MT-TSP than the ones from the MICP.

A Mixed-Integer Conic Program for the Moving-Target Traveling Salesman Problem based on a Graph of Convex Sets

Abstract

This paper introduces a new formulation that finds the optimum for the Moving-Target Traveling Salesman Problem (MT-TSP), which seeks to find a shortest path for an agent, that starts at a depot, visits a set of moving targets exactly once within their assigned time-windows, and returns to the depot. The formulation relies on the key idea that when the targets move along lines, their trajectories become convex sets within the space-time coordinate system. The problem then reduces to finding the shortest path within a graph of convex sets, subject to some speed constraints. We compare our formulation with the current state-of-the-art Mixed Integer Conic Program (MICP) solver for the MT-TSP. The experimental results show that our formulation outperforms the MICP for instances with up to 20 targets, with up to two orders of magnitude reduction in runtime, and up to a 60\% tighter optimality gap. We also show that the solution cost from the convex relaxation of our formulation provides significantly tighter lower bounds for the MT-TSP than the ones from the MICP.
Paper Structure (13 sections, 1 theorem, 8 equations, 4 figures, 2 tables)

This paper contains 13 sections, 1 theorem, 8 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. MICP for MT-TSP
  4. MICP on the Graph of Convex Sets
  5. Biconvex Binary Program for the MT-TSP
  6. GCS-Based Mixed Integer Conic Program (MICP-GCS)
  7. Proof of Validity
  8. Numerical Results
  9. Test Settings and Instance Generation
  10. Varying the Time-Window Duration
  11. Varying the Agent Speed
  12. Evaluating the Convex Relaxations
  13. Conclusion and Future Work

Key Result

Theorem 1

The optimal value of the MICP-GCS formulation is equal to the optimal value of the biconvex formulation for the MT-TSP. An optimal agent tour for the MT-TSP can be recovered from the solution of MICP-GCS by choosing $(p_i,t_i) \; \forall \; i \in V$ as shown in eq:recoverPti and eq:recoverPts'.

Figures (4)

  • Figure 1: A feasible solution to an example instance of the MT-TSP where 5 targets move along lines. The agent's tour is given in blue, and the part of each target's trajectory corresponding to its time-window where they can be visited by the agent are given by colored solid segments.
  • Figure 2: The trajectory-segment that corresponds to the time-window of some node $i \in V$, is a line segment within the space-time coordinate system $(x,y,t)$. The set of all points in the line segment forms the convex set $X_i$.
  • Figure 3: Numerical results comparing runtime and % Gap of the MICP and MICP-GCS for a fixed $v_{max}$ of 4, and varying time-window durations of 25 (a), 50 (b), and 75 (c). MICP-GCS scales significantly better than the MICP, when increasing the time-window duration, and number of targets. This can be seen especially for 15 targets in (b) and 10 targets in (c) where it runs up to 2 orders of magnitude faster while providing the same or better % Gap. Similarly, in the case of 15 targets in (c), and 20 targets in (b) and (c), MICP-GCS runs up to more than 1000 seconds faster, while providing a % Gap improvement within a 40-60 range.
  • Figure 4: Numerical results comparing runtime and % Gap of the MICP and MICP-GCS for a fixed time-window duration of 50, and varying $v_{max}$ choices of 6 (a), and 8 (b). The plots are very similar to the $v_{max}$ of 4 plot (Fig. \ref{['fig:exprTw']} (b)). Hence, here too, MICP-GCS gives two orders of magnitude faster runtime with a % Gap improvement of close to 10 for 15 targets. For 20 targets, MICP-GCS is still several hundreds of seconds faster, and gives a % Gap improvement of around 45.

Theorems & Definitions (2)

  • Theorem 1
  • proof