Editing Fitness function (section)

=== Comparison of both types of assessment ===
[[File:ParetoFront und Gewichtete Summe.png|left|thumb|Relationship between the Pareto front and the weighted sum. The set of feasible solutions <math>Z</math> is partially bounded by the Pareto front (green).<ref name=":1" />]]
[[File:ParetoFront nicht konvex.png|thumb|Example of a non-convex Pareto front<ref name=":1" />]]
With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is [[Convex set|convex]].<ref>{{Cite book |last=Miettinen |first=Kaisa |url=http://link.springer.com/10.1007/978-1-4615-5563-6 |title=Nonlinear Multiobjective Optimization |date=1998 |publisher=Springer US |isbn=978-1-4613-7544-9 |series=International Series in Operations Research & Management Science |volume=12 |location=Boston, MA |doi=10.1007/978-1-4615-5563-6}}</ref> This is illustrated by the adjacent picture on the left. The point <math>\mathsf{P}</math> on the green Pareto front is reached by the weights <math>w_1</math> and <math>w_2</math>, provided that the EA converges to the optimum. The direction with the largest fitness gain in the solution set <math>Z</math> is shown by the drawn arrows.

In case of a non-convex front, however, non-convex front sections are not reachable by the weighted sum. In the adjacent image on the right, this is the section between points <math>\mathsf{A}</math> and <math>\mathsf{B}</math>. This can be remedied to a limited extent by using an extension of the weighted sum, the ''cascaded weighted sum''.<ref name=":1" />

Comparing both assessment approaches, the use of Pareto optimization is certainly advantageous when little is known about the possible solutions of a task and when the number of optimization objectives can be narrowed down to three, at most four. However, in the case of repeated optimization of variations of one and the same task, the desired lines of compromise are usually known and the effort to determine the entire Pareto front is no longer justified. This is also true when no human decision is desired or possible after optimization, such as in automated decision processes.<ref name=":1" />