Error-Bounded Approximation of Pareto Fronts in Robot Planning Problems

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.

Online Multi-Robot Task Assignment with Stochastic Blockages

“…Many robotic planning problems face the challenge of being required to simultaneously optimize multiple objectives, for instance in path and trajectory planning [5], [8], [10], autonomous driving [11]- [14] transportation and mobility on demand [3], multi-robot planning [15]- [19].…”

“…A common approach to multi-objective optimization is linear scalarization, i.e., using the weighted sum of the individual objective functions to pose a single optimization problem [9]. The set of Pareto-optimal solutions is then approximated by exploring different scalarization weights [5], [8], [28]- [30]. However, finding useful weights is often challenging [8], [9], [31], [32].…”

“…Thus, learning a user's reward function is equivalent to finding the Pareto-optimal trade-off that best fit their preferences. Furthermore, effectively computing a representative set of Pareto-optimal solutions improves how well system behaviour can be adapted to user preferences [8]. In our earlier work, we studied the problem of learning user preferences for material transport with consideration of task efficiency and following social norms [5], [56].…”

“…A commonly used approach to find different trade-offs of cost functions is sampling weights uniformly. However, the corresponding solutions are often not placed uniformly on the Pareto front since the mapping from weights to the objective values is non-linear [8]. This can lead to several weights yielding similar objective values.…”

“…Each possible choice of scalarization weights then corresponds to an MPRD policy. However, picking weights that lead to a desirable set of policies is not trivial, since the relationship between weights and MRPD costs is often non-linear [8], [9]. Picking regularly spaced weights can lead to several policies having similar behaviours, i.e., trade-offs, while other possible system behaviours are not covered by any policy.…”