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
Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making that is concerned with ...
problems (MOP) are given.
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, Haupt et al. and from Rody Oldenhuis software. 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,
[Deb, Kalyanmoy (2002) Multiobjective optimization using evolutionary algorithms (Repr. ed.). Chichester .a. Wiley. .] Binh et al.
[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 and Binh.
[Binh T. (1999]
A multiobjective evolutionary algorithm. The study cases.
Technical report. Institute for Automation and Communication. Barleben, Germany The software developed by Deb can be downloaded,
[Deb K. (2011) Software for multi-objective NSGA-II code in C. Available at URL: https://www.iitk.ac.in/kangal/codes.shtml] which implements the NSGA-II procedure with GAs, or the program posted on Internet, 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.
Test functions for single-objective optimization
Test functions for constrained optimization
Test functions for multi-objective optimization
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Binh and Korn function:
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Chankong and Haimes function:
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Kursawe function:
[F. Kursawe, ]
A variant of evolution strategies for vector optimization
” in PPSN I, Vol 496 Lect Notes in Comput Sc. Springer-Verlag, 1991, pp. 193–197.
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, Schaffer function N. 1:
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, Poloni's two objective function:
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, Osyczka and Kundu function:
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, CTP1 function (2 variables):
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, Constr-Ex problem:
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See also
*
Ackley function
*
Himmelblau's function
In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by:
: f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2.\quad
It has one local maximum at x = -0 ...
*
Rastrigin function
*
Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non- convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Ros ...
*
Shekel function
*
Binh function
References
{{DEFAULTSORT:Test functions for optimization
Constraint programming
Convex optimization
Types of functions