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Regional Constellation Reconfiguration Problem: Integer Linear Programming Formulation and Lagrangian Heuristic Method

Hang Woon Lee, Koki Ho

Abstract

A group of satellites, with either homogeneous or heterogeneous orbital characteristics and/or hardware specifications, can undertake a reconfiguration process due to variations in operations pertaining to Earth observation missions. This paper investigates the problem of optimizing a satellite constellation reconfiguration process against two competing mission objectives: (i) the maximization of the total coverage reward and (ii) the minimization of the total cost of the transfer. The decision variables for the reconfiguration process include the design of the new configuration and the assignment of satellites from one configuration to another. We present a novel bi-objective integer linear programming formulation that combines constellation design and transfer problems. The formulation lends itself to the use of generic mixed-integer linear programming (MILP) methods such as the branch-and-bound algorithm for the computation of provably-optimal solutions; however, these approaches become computationally prohibitive even for moderately-sized instances. In response to this challenge, this paper proposes a Lagrangian relaxation-based heuristic method that leverages the assignment problem structure embedded in the problem. The results from the computational experiments attest to the near-optimality of the Lagrangian heuristic solutions and a significant improvement in the computational runtime compared to a commercial MILP solver.

Regional Constellation Reconfiguration Problem: Integer Linear Programming Formulation and Lagrangian Heuristic Method

Abstract

A group of satellites, with either homogeneous or heterogeneous orbital characteristics and/or hardware specifications, can undertake a reconfiguration process due to variations in operations pertaining to Earth observation missions. This paper investigates the problem of optimizing a satellite constellation reconfiguration process against two competing mission objectives: (i) the maximization of the total coverage reward and (ii) the minimization of the total cost of the transfer. The decision variables for the reconfiguration process include the design of the new configuration and the assignment of satellites from one configuration to another. We present a novel bi-objective integer linear programming formulation that combines constellation design and transfer problems. The formulation lends itself to the use of generic mixed-integer linear programming (MILP) methods such as the branch-and-bound algorithm for the computation of provably-optimal solutions; however, these approaches become computationally prohibitive even for moderately-sized instances. In response to this challenge, this paper proposes a Lagrangian relaxation-based heuristic method that leverages the assignment problem structure embedded in the problem. The results from the computational experiments attest to the near-optimality of the Lagrangian heuristic solutions and a significant improvement in the computational runtime compared to a commercial MILP solver.
Paper Structure (25 sections, 1 theorem, 29 equations, 10 figures, 4 tables, 3 algorithms)

This paper contains 25 sections, 1 theorem, 29 equations, 10 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Constellation Design and Transfer Problems
  3. Constellation-Coverage Model
  4. Model Definitions
  5. Special Case: APC Decomposition
  6. Constellation Design: Maximum Coverage Problem
  7. Constellation Transfer: Assignment Problem
  8. Regional Constellation Reconfiguration Problem
  9. Problem Description
  10. Mathematical Formulation
  11. Model Characteristics
  12. Lagrangian Relaxation-Based Solution Method
  13. $\varepsilon$-constraint Reformulation
  14. Lagrangian Relaxation
  15. Lagrangian Dual Problem and Subgradient Method
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\bm{A}$ be an integral matrix. The polyhedron $\{\bm{x}:\bm{A}\bm{x}\le\bm{b},\bm{x}\ge \bm{0}\}$ is integral for all integral vector $\bm{b}$ if and only if $\bm{A}$ is TU.

Figures (10)

  • Figure 1: MCP solution for Example \ref{['eg:3']}.
  • Figure 2: Decision variables of AP and MCP and their relationship.
  • Figure 3: Illustration of the 1-exchange operation.
  • Figure 4: Computational results for instances 1--9. Note that all metrics are normalized and flipped in sign.
  • Figure 5: Computational results for instances 10--18. Note that all metrics are normalized and flipped in sign.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Definition 1: Visibility matrix
  • Definition 2: Constellation pattern vector
  • Definition 3: Coverage timeline
  • Remark 1: Linear property
  • Definition 4: Reference visibility profile
  • Definition 5: Visibility circulant matrix
  • Remark 2: Circular convolution operation lee2020satellite
  • Example 1: 5-satellite MCP
  • Definition 6: Total unimodularity
  • Theorem 1: Hoffman-Kruskal hoffman1956
  • ...and 1 more