Test functions for optimization: Difference between revisions

Revision as of 07:24, 22 October 2020

Revision as of 02:05, 10 December 2020

Line 9:

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=978-0-19-509971-3|page=328}}</ref> Haupt et al.<ref>{{cite book|last=Haupt|first=Randy L. Haupt, Sue Ellen|title=Practical genetic algorithms with CD-Rom|year=2004|publisher=J. Wiley|___location=New York|isbn=978-0-471-45565-3|edition=2nd}}</ref> and from Rody Oldenhuis software.<ref>{{cite web|last=Oldenhuis|first=Rody|title=Many test functions for global optimizers|url=http://www.mathworks.com/matlabcentral/fileexchange/23147-many-testfunctions-for-global-optimizers|publisher=Mathworks|accessdateaccess-date=1 November 2012}}</ref> Given the number of problems (55 in total), just a few are presented here. The complete list of test functions is found on the Mathworks website.<ref>{{cite web|last=Ortiz|first=Gilberto A.|title=Evolution Strategies (ES)|url=http://www.mathworks.com/matlabcentral/fileexchange/35801-evolution-strategies-es|publisher=Mathworks|accessdateaccess-date=1 November 2012}}</ref>

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) [https://pdfs.semanticscholar.org/cf68/41a6848ca2023342519b0e0e536b88bdea1d.pdf 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: https://www.iitk.ac.in/kangal/codes.shtml</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|accessdateaccess-date=1 November 2012}}</ref> 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.

Line 185:

! Name !! Plot !! Formula !! Global minimum !! Search ___domain

|-

| Rosenbrock function constrained with a cubic and a line<ref>{{cite conference |author1=Simionescu, P.A. |author2=Beale, D. |title=New Concepts in Graphic Visualization of Objective Functions |conference=ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference |___location=Montreal, Canada |date=September 29 – October 2, 2002|pages=891–897 |url=http://faculty.tamucc.edu/psimionescu/PDFs/DETC02-DAC-34129.pdf |accessdateaccess-date=7 January 2017 }}</ref>

|| [[File:ConstrTestFunc04.png|200px|Rosenbrock function constrained with a cubic and a line]]

|| <math>f(x,y) = (1-x)^2 + 100(y-x^2)^2</math>,

Line 201:

|| <math>-1.5\le x \le 1.5</math>, <math>-1.5\le y \le 1.5</math>

|-

| 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|access-date=2017-08-29|url-status=bot: unknown|archiveurlarchive-url=https://web.archive.org/web/20161229032528/http://www.phoenix-int.com/software/benchmark_report/bird_constrained.php|archivedatearchive-date=2016-12-29}}</ref><ref>{{Cite journal|last=Mishra|first=Sudhanshu|date=2006|title=Some new test functions for global optimization and performance of repulsive particle swarm method|url=https://mpra.ub.uni-muenchen.de/2718/|journal=MPRA Paper|volume=|pages=|via=}}</ref>

|| [[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>,