Editing Fitness function (section)

=== Pareto optimization ===
A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives <math>f_1</math> and <math>f_2</math> to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution.<ref name=":0" /> Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides.<ref name=":2">{{Cite book |last=Deb |first=Kalyanmoy |url=https://www.researchgate.net/publication/242505658 |title=Multiobjective Optimization: Interactive and Evolutionary Approaches |date=2008 |publisher=Springer |isbn=978-3-540-88907-6 |editor-last=Branke |editor-first=Jürgen |series=Lecture Notes in Computer Science |volume=5252 |location=Berlin, Heidelberg |pages=58–96 |language=en |chapter=Introduction to Evolutionary Multiobjective Optimization |doi=10.1007/978-3-540-88908-3 |editor-last2=Deb |editor-first2=Kalyanmoy |editor-last3=Miettinen |editor-first3=Kaisa |editor-last4=Słowiński |editor-first4=Roman |s2cid=15375227}}</ref>

It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well.<ref name=":2" /><ref>{{Cite journal |last1=Fonseca |first1=Carlos M. |last2=Fleming |first2=Peter J. |date=1995 |title=An Overview of Evolutionary Algorithms in Multiobjective Optimization |url=https://direct.mit.edu/evco/article/3/1/1-16/733 |journal=Evolutionary Computation |language=en |volume=3 |issue=1 |pages=1–16 |doi=10.1162/evco.1995.3.1.1 |s2cid=8530790 |issn=1063-6560|url-access=subscription }}</ref> They are therefore well suited as [[Multi-objective optimization#A posteriori methods|a-posteriori methods]] for multi-objective optimization, in which the final decision is made by a human decision maker after optimization and determination of the Pareto front.<ref name=":0" /> Besides the SPEA2,<ref>{{Cite journal |last1=Eckart |first1=Zitzler |last2=Marco |first2=Laumanns |last3=Lothar |first3=Thiele |date=2001 |title=SPEA2: Improving the strength pareto evolutionary algorithm |url= |journal=Technical Report, Nr. 103. Computer Engineering and Networks Laboratory (TIK) |language=en |publisher=ETH Zürich 2001 |pages= |doi=10.3929/ethz-a-004284029|s2cid=16584254 }}</ref> the NSGA-II<ref>{{Cite journal |last1=Deb |first1=K. |last2=Pratap |first2=A. |last3=Agarwal |first3=S. |last4=Meyarivan |first4=T. |date=2002 |title=A fast and elitist multiobjective genetic algorithm: NSGA-II |url=https://ieeexplore.ieee.org/document/996017 |journal=IEEE Transactions on Evolutionary Computation |volume=6 |issue=2 |pages=182–197 |doi=10.1109/4235.996017|s2cid=9914171 }}</ref> and NSGA-III<ref>{{Cite journal |last1=Deb |first1=Kalyanmoy |last2=Jain |first2=Himanshu |date=2014 |title=An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints |url=https://ieeexplore.ieee.org/document/6600851 |journal=IEEE Transactions on Evolutionary Computation |volume=18 |issue=4 |pages=577–601 |doi=10.1109/TEVC.2013.2281535 |s2cid=206682597 |issn=1089-778X|url-access=subscription }}</ref><ref>{{Cite journal |last1=Jain |first1=Himanshu |last2=Deb |first2=Kalyanmoy |date=2014 |title=An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach |url=https://ieeexplore.ieee.org/document/6595567 |journal=IEEE Transactions on Evolutionary Computation |volume=18 |issue=4 |pages=602–622 |doi=10.1109/TEVC.2013.2281534 |s2cid=16426862 |issn=1089-778X|url-access=subscription }}</ref> have established themselves as standard methods.

The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives.<ref name=":1">{{Cite journal |last1=Jakob |first1=Wilfried |last2=Blume |first2=Christian |date=2014-03-21 |title=Pareto Optimization or Cascaded Weighted Sum: A Comparison of Concepts |journal=Algorithms |language=en |volume=7 |issue=1 |pages=166–185 |arxiv=2203.02697 |doi=10.3390/a7010166 |issn=1999-4893|doi-access=free }}</ref> If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods.<ref name=":0" />