Many optimization problems have multiple competing objectives. These competing objectives are part of the trade-off that defines an optimal solution. Sometimes these competing objectives have separate priorities where one objective should be satisfied before another objective is even considered. This especially arises in model predictive control or other types of dynamic optimization problems. There are competing objectives with a ranked hierarchy. The highest level objectives are satisfied first followed by lower ranked objectives if there are additional degrees of freedom available. The l1-norm objective is a natural way to explicitly rank objectives and simultaneously optimize multiple priorities with a single optimization problem.
Consider examples of safety, environmental, and economic constraints or objectives. Which are most important and why?
For the following multi-objective optimization problem, sketch a possible optimal trajectory.