![](bg4.png)
visited by j-th individual in all iterations and C
j
is
the coefficient effect of i-th krill individual which is
the exponentially distributed random number. The
larger random numbers are allocated to individuals
with higher fitness and the smaller ones are allo-
cated to the individuals with lower fitness. Also K
j,ibest
equals to best fitness of j-th individual in all it-
erations and K
i
equals to fitness of i-th individual, K
best
and K
worst
are the best and worst fitness achieved
by all krill individuals.
4 EXPERIMENTATION
In order to evaluate the performance of the proposed
algorithms, we applied it to a set of standard bench-
mark functions listed in Table 1. These benchmarks
include high dimensional functions which are diffi-
cult to solve due their dimension (Fister et al., 2013)
and are being used most frequently by the researchers
to examine the performance of the different optimiza-
tion algorithms. If we exclude Sphere and DixonPrice
which are unimodal functions, the rest are multimodal
functions. In spite of unimodal functions which have
one local optimum, multimodal functions have many
local minimum points with a risk of being trapped in
them. In Table 1, n is the dimension of the functions,
Search Space is the problem space which is a subset
of R
n
. The Global Minimum is the minimum value
of the functions which are zero for all functions ex-
cept for Michalewicz function (Its minimum point is
-9.66). The dimension for F1 to F9 functions is con-
sidered 20 and for F10 is considered 10. Below is the
description of the benchmark functions we have used
in our experiments. More details can be found in (Ali
et al., 2005).
F1. Ackley function: This is a popular test problem
for evaluating the performance of the optimization
methods. Its many local optimum solutions chal-
lenge the performance of the optimization meth-
ods by posing a risk on them, to be trapped in one
of local optimum solutions and this is specially
the case for the hill climbing methods. Ackley
is continuous, differentiable, non-separable, scal-
able and multimodal. The global minimum of the
function is f (x
?
) = 0 with corresponding solution
x
?
= (0,...,0). The test domain is 32.768 ≤ x
i
≤
32.768.
F2. Griewank function: This function has many local
optimum solutions that are regularly distributed in
the problem space. This is also a non-separable,
scalable and a differentiable function. Its global
optimum solution is f (x
?
) = 0 with correspond-
ing solution x
?
= (0, . . . , 0). The domain of test is
600 ≤ x
i
≤ 600.
F3. Levy function: Levy is a continuous optimiza-
tion problem with several local optimum solution
distributed in the problem space. It has global op-
timum solution f (x
?
) = 0 which is located at x
?
=
(1,...,1). This problem is subject to 10 ≤x
i
≤10.
F4. Rastrigin function: Rastrigin is a continuous
multimodal function with many local optimum
solution distributed in the search space. It is a
difficult problem to solve due to its large search
space and large number of local optimum solu-
tions. Its global optimum solution is f (x
?
) = 0
with corresponding zero vector x
?
= (0, . . . , 0).
The test domain is 5.12 ≤ x
i
≤ 5.12.
F5. Schwefel function: Schwefel belongs to contin-
uous multimodal class of test functions. It is also
differ-entiable, separable and scalable functions.
Its many local optimum solutions make it gener-
ally difficult solution to solve. Its global minimum
is f (x
?
) = 0 which is located at x
?
= (0, . . . , 0).
The problem constraint is 500 ≤ x
i
≤ 500.
F6. Dixon Price function: This function is continu-
ous, differentiable, non-separable and unimodal
function. It has global minimum f (x
?
) = 0 which
is located at x
?
= (2(
2
i
−2
2
i
)) for i = 1 ...n, where n
is the dimension of the problem. The test domain
is 10 ≤ x
i
≤ 10.
F7. Rosenbrock function: Rosenbrock is a popu-
lar function for gradient-based optimization al-
gorithms. It is continuous, differentiable, non-
separable and unimodal function. It has global
minimum f (x
?
) = 0 which is located in nar-
row valley. The corresponding solution is x
?
=
(1,...,1) and the problem constraint is 5 ≤ x
i
≤
10.
F8. Sphere function: Sphere is a poplar test function
which is used most frequently by the researchers
for examining the performance of the optimiza-
tion methods. This function is continuous, differ-
entiable, separable and unimodal test function. Its
global optimum solution is f (x
?
) = 0 with corre-
sponding solution x
?
= (0,...,0), where 5.12 ≤
x
i
≤ 5.12.
F9. Powell function: Powell function is continuous,
differentiable, separable and unimodal function.
It has global optimum f (x
?
) = 0 which located
at x
?
= (3, 1, 0, 1 . . . , 3, 1, 0, 1) where 4 ≤ x
i
≤ 5.
F10. Michalewicz function: This function is contin-
uous multimodal function. It has global minimum
f (x
?
) = 9.66 for 10 dimension version (n = 10).
This problem is subject to 0 ≤ x
i
≤ π.
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