  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.
