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Efficient Multi-Objective Planning with Weighted Maximization Using Large Neighbourhood Search

Krishna Kalavadia, Shamak Dutta, Yash Vardhan Pant, Stephen L. Smith

Abstract

Autonomous navigation often requires the simultaneous optimization of multiple objectives. The most common approach scalarizes these into a single cost function using a weighted sum, but this method is unable to find all possible trade-offs and can therefore miss critical solutions. An alternative, the weighted maximum of objectives, can find all Pareto-optimal solutions, including those in non-convex regions of the trade-off space that weighted sum methods cannot find. However, the increased computational complexity of finding weighted maximum solutions in the discrete domain has limited its practical use. To address this challenge, we propose a novel search algorithm based on the Large Neighbourhood Search framework that efficiently solves the weighted maximum planning problem. Through extensive simulations, we demonstrate that our algorithm achieves comparable solution quality to existing weighted maximum planners with a runtime improvement of 1-2 orders of magnitude, making it a viable option for autonomous navigation.

Efficient Multi-Objective Planning with Weighted Maximization Using Large Neighbourhood Search

Abstract

Autonomous navigation often requires the simultaneous optimization of multiple objectives. The most common approach scalarizes these into a single cost function using a weighted sum, but this method is unable to find all possible trade-offs and can therefore miss critical solutions. An alternative, the weighted maximum of objectives, can find all Pareto-optimal solutions, including those in non-convex regions of the trade-off space that weighted sum methods cannot find. However, the increased computational complexity of finding weighted maximum solutions in the discrete domain has limited its practical use. To address this challenge, we propose a novel search algorithm based on the Large Neighbourhood Search framework that efficiently solves the weighted maximum planning problem. Through extensive simulations, we demonstrate that our algorithm achieves comparable solution quality to existing weighted maximum planners with a runtime improvement of 1-2 orders of magnitude, making it a viable option for autonomous navigation.

Paper Structure

This paper contains 28 sections, 4 theorems, 15 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Problem Formulation
  5. Linking Weighted Maximization and Summation
  6. Limitations of the Weighted Sum
  7. On the Convexity of Subproblem Pareto Fronts
  8. Large Neighbourhood Search for Weighted Maximization
  9. WM-LNS Solver Framework
  10. Initial Solution Generation
  11. Destroy Procedure
  12. Worst Removal
  13. Best Removal
  14. Unbalanced Objective Removal
  15. Balanced Objective Removal
  16. ...and 13 more sections

Key Result

Lemma 1

Let $\mathcal{P}_{\text{SPO}} \subseteq \mathcal{P}_{s,g}$ be the set of all paths whose cost vectors are non-dominated and lie on the boundary of the convex hull of $\mathcal{F}_{s,g}$. Then, $\mathcal{P}_{\text{WS}} = \mathcal{P}_{\text{SPO}}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: Solving multi-objective path planning problems via scalarization. An external decision maker specifies objective preferences as weights $w_1, w_2$. The path planner solves the scalarized problem and returns a path for the robot to execute. The WS cannot return the solution matching the desired objective preferences because it lies in the non-convex region of the Pareto front.
  • Figure 2: An example of the changing geometry of the Pareto front depending on the subproblem considered. The global Pareto front corresponds to optimizing over $\mathcal{P}_{s, g}$. The local Pareto front corresponds to the problem of optimizing over $\mathcal{P}_{v_i, v_j}$.
  • Figure 3: Solution sets obtained when optimizing two objectives: path length and obstacle closeness. The upper row shows the solution sets found by the WS, the middle row shows the solutions found by the WM, and the bottom row shows the solutions found by the proposed WM-LNS solver. The left column shows the paths found, and the right column shows the corresponding objective values. Note that the WM-LNS plots include suboptimal solutions for illustration; however, Table \ref{['tab:solution_set_summary']} reports only Pareto-optimal solutions.
  • Figure 4: Performance of WM-LNS over two instances. Left plots show the computation time ratio over WS, and right plots show the percentage error from the optimal WM solution.
  • Figure 5: Test maps used in our experiments. Left: Map 1 - Maze (Two Objectives). Right: Map 2 - Cluttered Boxes (Three Objectives). Red regions correspond to high-risk zones, and green regions correspond to low-risk zones. The paths illustrate solutions for balanced preferences, where weights are chosen such that the weighted objective values are approximately equal.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: Pareto-optimality
  • Lemma 1
  • Proposition 1: Proposition 1, wilde_scalarizing_2024
  • Proposition 2: Approximation Bound
  • proof
  • Proposition 3: Hardness of BWSA
  • proof
  • Remark 1: Hardness for a fixed number of objectives