Error-Bounded Approximation of Pareto Fronts in Robot Planning Problems

TL;DR

This paper introduces an error-bounded method for approximating Pareto fronts in multi-objective robot planning, providing efficient weight vector selection with proven properties and superior performance over uniform sampling.

Contribution

It presents a novel algorithm that greedily selects weight vectors with error bounds, leveraging properties of the optimal cost function for better Pareto front approximation.

Findings

Outperforms uniform weight distribution in robot planning tasks

Provides theoretical bounds on approximation error

Demonstrates effectiveness across multiple objectives

Abstract

Many problems in robotics seek to simultaneously optimize several competing objectives under constraints. A conventional approach to solving such multi-objective optimization problems is to create a single cost function comprised of the weighted sum of the individual objectives. Solutions to this scalarized optimization problem are Pareto optimal solutions to the original multi-objective problem. However, finding an accurate representation of a Pareto front remains an important challenge. Using uniformly spaced weight vectors is often inefficient and does not provide error bounds. Thus, we address the problem of computing a finite set of weight vectors such that for any other weight vector, there exists an element in the set whose error compared to optimal is minimized. To this end, we prove fundamental properties of the optimal cost as a function of the weight vector, including its continuity and concavity. Using these, we propose an algorithm that greedily adds the weight vector least-represented by the current set, and provide bounds on the error. Finally, we illustrate that the proposed approach significantly outperforms uniformly distributed weights for different robot planning problems with varying numbers of objective functions.