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→Test functions for single-objective optimization: Citation for certified global minima of egg holder function |
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Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for [[multi-objective optimization]] problems (MOP) are given.
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck,<ref>{{cite book|last=Bäck|first=Thomas|title=Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms|year=1995|publisher=Oxford University Press|___location=Oxford|isbn=978-0-19-509971-
The test functions used to evaluate the algorithms for MOP were taken from Deb,<ref name="Deb:2002">Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. {{isbn|0-471-87339-X}}.</ref> Binh et al.<ref name="Binh97">Binh T. and Korn U. (1997) MOBES: A Multiobjective Evolution Strategy for Constrained Optimization Problems. In: Proceedings of the Third International Conference on Genetic Algorithms. Czech Republic. pp. 176–182</ref> and Binh.<ref name="Binh99">Binh T. (1999) [https://www.researchgate.net/profile/Thanh_Binh_To/publication/2446107_A_Multiobjective_Evolutionary_Algorithm_The_Study_Cases/links/53eb422f0cf28f342f45251d.pdf A multiobjective evolutionary algorithm. The study cases.] Technical report. Institute for Automation and Communication. Barleben, Germany</ref> You can download the software developed by Deb,<ref name="Deb_nsga">Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL:http://www.iitk.ac.in/kangal/codes.shtml. Revision 1.1.6</ref> which implements the NSGA-II procedure with GAs, or the program posted on Internet,<ref>{{cite web|last=Ortiz|first=Gilberto A.|title=Multi-objective optimization using ES as Evolutionary Algorithm.|url=http://www.mathworks.com/matlabcentral/fileexchange/35824-multi-objective-optimization-using-evolution-strategies-es-as-evolutionary-algorithm-ea|publisher=Mathworks|accessdate=1 November 2012}}</ref> which implements the NSGA-II procedure with ES.
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|| <math>-1.5\le x \le 1.5</math>, <math>-1.5\le y \le 1.5</math>
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| Mishra's Bird function - constrained<ref>{{Cite web|url=http://www.phoenix-int.com/software/benchmark_report/bird_constrained.php|title=Bird Problem (Constrained) {{!}} Phoenix Integration
|| [[File:ConstrTestFunc01.png|200px|Bird function (constrained)]]
|| <math>f(x,y) = \sin(y) e^{\left [(1-\cos x)^2\right]} + \cos(x) e^{\left [(1-\sin y)^2 \right]} + (x-y)^2</math>,
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|| <math>-20\le x,y \le 20</math>
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| [[Fonseca–Fleming function]]:<ref name="FonzecaFleming:1995">{{cite journal |first=C. M. |last=Fonseca |first2=P. J. |last2=Fleming |title=An Overview of Evolutionary Algorithms in Multiobjective Optimization |journal=[[Evolutionary Computation (journal)|Evol Comput]] |volume=3 |issue=1 |pages=1–16 |year=1995 |doi=10.1162/evco.1995.3.1.1 |citeseerx=10.1.1.50.7779 }}</ref>
|| [[File:Fonseca and Fleming function.pdf|200px|Fonseca and Fleming function]]
|| <math>\text{Minimize} =
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