Test functions for optimization

In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as: where:  A = 10 {displaystyle { ext{where: }}A=10} − exp ⁡ [ 0.5 ( cos ⁡ 2 π x + cos ⁡ 2 π y ) ] + e + 20 {displaystyle -exp left+e+20} + ( 2.625 − x + x y 3 ) 2 {displaystyle +left(2.625-x+xy^{3} ight)^{2}} [ 30 + ( 2 x − 3 y ) 2 ( 18 − 32 x + 12 x 2 + 48 y − 36 x y + 27 y 2 ) ] {displaystyle left} + ( y − 1 ) 2 ( 1 + sin 2 ⁡ 2 π y ) {displaystyle +left(y-1 ight)^{2}left(1+sin ^{2}2pi y ight)} subjected to: ( x − 1 ) 3 − y + 1 ≤ 0  and  x + y − 2 ≤ 0 {displaystyle (x-1)^{3}-y+1leq 0{ ext{ and }}x+y-2leq 0} subjected to: x 2 + y 2 ≤ 2 {displaystyle x^{2}+y^{2}leq 2} subjected to: ( x + 5 ) 2 + ( y + 5 ) 2 < 25 {displaystyle (x+5)^{2}+(y+5)^{2}<25} subjected to: x 2 + y 2 < [ 2 cos ⁡ t − 1 2 cos ⁡ 2 t − 1 4 cos ⁡ 3 t − 1 8 cos ⁡ 4 t ] 2 + [ 2 sin ⁡ t ] 2 {displaystyle x^{2}+y^{2}<left^{2}+^{2}} where: t = Atan2(x,y)subjected to: x 2 + y 2 ≤ [ r T + r S cos ⁡ ( n arctan ⁡ x y ) ] 2 {displaystyle x^{2}+y^{2}leq left^{2}} where:  r T = 1 , r S = 0.2  and  n = 8 {displaystyle { ext{where: }}r_{T}=1,r_{S}=0.2{ ext{ and }}n=8} where = { A 1 = 0.5 sin ⁡ ( 1 ) − 2 cos ⁡ ( 1 ) + sin ⁡ ( 2 ) − 1.5 cos ⁡ ( 2 ) A 2 = 1.5 sin ⁡ ( 1 ) − cos ⁡ ( 1 ) + 2 sin ⁡ ( 2 ) − 0.5 cos ⁡ ( 2 ) B 1 ( x , y ) = 0.5 sin ⁡ ( x ) − 2 cos ⁡ ( x ) + sin ⁡ ( y ) − 1.5 cos ⁡ ( y ) B 2 ( x , y ) = 1.5 sin ⁡ ( x ) − cos ⁡ ( x ) + 2 sin ⁡ ( y ) − 0.5 cos ⁡ ( y ) {displaystyle { ext{where}}={egin{cases}A_{1}=0.5sin left(1 ight)-2cos left(1 ight)+sin left(2 ight)-1.5cos left(2 ight)\A_{2}=1.5sin left(1 ight)-cos left(1 ight)+2sin left(2 ight)-0.5cos left(2 ight)\B_{1}left(x,y ight)=0.5sin left(x ight)-2cos left(x ight)+sin left(y ight)-1.5cos left(y ight)\B_{2}left(x,y ight)=1.5sin left(x ight)-cos left(x ight)+2sin left(y ight)-0.5cos left(y ight)end{cases}}} 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 (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 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.