Scalarization

For the motion planning problem, a Pareto-optimal solution is also optimal for a scalar cost functional that is constructed as a linear combination of the individual costs. Let $\alpha_1$, $\ldots$, $\alpha_m$ be positive real constants, and let

$$l(\gamma) = \sum_{i=1}^m \alpha_i L_i(\gamma) .$$ (7.29)

It is easy to show that any plan that is optimal with respect to (7.29) is also a Pareto-optimal solution [606]. If a Pareto optimal solution is generated in this way, however, there is no easy way to determine what alternatives exist.