Test functions for optimization

In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:

Convergence rate.
Precision.
Robustness.
General performance.

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,^[1] Haupt et al.^[2] and from Rody Oldenhuis software.^[3] Given the number of problems (55 in total), just a few are presented here.

The test functions used to evaluate the algorithms for MOP were taken from Deb,^[4] Binh et al.^[5] and Binh.^[6] The software developed by Deb can be downloaded,^[7] which implements the NSGA-II procedure with GAs, or the program posted on Internet,^[8] which implements the NSGA-II procedure with ES.

Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein.

^ Bäck, Thomas (1995). Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms. Oxford: Oxford University Press. p. 328. ISBN 978-0-19-509971-3.
^ Haupt, Randy L. Haupt, Sue Ellen (2004). Practical genetic algorithms with CD-Rom (2nd ed.). New York: J. Wiley. ISBN 978-0-471-45565-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Oldenhuis, Rody. "Many test functions for global optimizers". Mathworks. Retrieved 1 November 2012.
^ Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. ISBN 0-471-87339-X.
^ 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
^ Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany
^ Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml
^ Ortiz, Gilberto A. "Multi-objective optimization using ES as Evolutionary Algorithm". Mathworks. Retrieved 1 November 2012.

[1] Bäck, Thomas (1995). Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms. Oxford: Oxford University Press. p. 328. ISBN 978-0-19-509971-3.

[2] Haupt, Randy L. Haupt, Sue Ellen (2004). Practical genetic algorithms with CD-Rom (2nd ed.). New York: J. Wiley. ISBN 978-0-471-45565-3.{{cite book}}: CS1 maint: multiple names: authors list (link)

[3] Oldenhuis, Rody. "Many test functions for global optimizers". Mathworks. Retrieved 1 November 2012.

[Deb:2002-4] Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester [u.a.]: Wiley. ISBN 0-471-87339-X.

[Binh97-5] 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

[Binh99-6] Binh T. (1999) A multiobjective evolutionary algorithm. The study cases. Technical report. Institute for Automation and Communication. Barleben, Germany

[Deb_nsga-7] Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml

[8] Ortiz, Gilberto A. "Multi-objective optimization using ES as Evolutionary Algorithm". Mathworks. Retrieved 1 November 2012.

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