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.

翻译：机器人的许多问题都试图同时在制约下优化若干相互竞争的目标。解决这种多目标优化问题的常规方法是创建一个单一的成本函数,由单个目标的加权和总和组成。这个升级优化问题的解决方案是最初的多目标问题的最佳解决方案。然而,找到对帕雷托前方的准确表述仍是一个重大挑战。使用统一空间重量矢量往往效率低下,而且不提供错误界限。因此,我们处理计算一定的一组重量矢量的问题,例如,对于任何其他重量矢量而言,在集中存在一个其误差与最佳误差最小化的元素。为此,我们证明最佳成本作为重量矢量的函数的基本特性,包括其连续性和共性。我们使用这些算法,提出一种贪婪地增加当前设定中代表最少的重量矢量的算法,并提供误差的界限。最后,我们指出,拟议的方法明显超出了具有不同数量客观功能的不同机器人规划问题的统一分布重量。