Test functions for optimization


In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:
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 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, just a few are presented here. The complete list of test functions is found on the Mathworks website.
The test functions used to evaluate the algorithms for MOP were taken from Deb, Binh et al. and Binh. You can download the software developed by Deb, 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

NamePlotFormulaGlobal minimumSearch domain
Rastrigin function
Ackley function
Sphere function,
Rosenbrock function,
Beale function
Goldstein–Price function
Booth function
Bukin function N.6,
Matyas function
Lévi function N.13
Himmelblau's function
Three-hump camel function
Easom function
Cross-in-tray function
Eggholder function
Hölder table function
McCormick function,
Schaffer function N. 2
Schaffer function N. 4
Styblinski–Tang function,..

Test functions for constrained optimization

NamePlotFormulaGlobal minimumSearch domain
Rosenbrock function constrained with a cubic and a line,
subjected to:		,
Rosenbrock function constrained to a disk,
subjected to:		,
Mishra's Bird function - constrained,
subjected to:		,
Townsend function ,
subjected to:
where: 		,
Simionescu function,
subjected to:

Test functions for multi-objective optimization

NamePlotFunctionsConstraintsSearch domain
Binh and Korn function:,
Chankong and Haimes function:
Fonseca–Fleming function:,
Test function 4:
Kursawe function:,.
Schaffer function N. 1:. Values of from to have been used successfully. Higher values of increase the difficulty of the problem.
Schaffer function N. 2:.
Poloni's two objective function:
Zitzler–Deb–Thiele's function N. 1:,.
Zitzler–Deb–Thiele's function N. 2:,.
Zitzler–Deb–Thiele's function N. 3:,.
Zitzler–Deb–Thiele's function N. 4:,,
Zitzler–Deb–Thiele's function N. 6:,.
Osyczka and Kundu function:,,.
CTP1 function :.
Constr-Ex problem:,
Viennet function:.