Harmony Search as a Metaheuristic Algorithm - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 5

For the first benchmark example, we test Rosenbrock's logarithmic banana func-
tion as follows:

2 2 2

f ( x , y ) = ln[ 1 + ( − x ) + 100 ( y − x ) ] (6)

where ( x , y )∈[−10, 10]×[−10, 10] and the global minimum f min =0 at (1, 1). The HS
algorithm found the global optimum successfully. The variations of these solutions
and their paths are shown in Figure 2.
We have used 20 harmonies with harmony accepting rate r accept =0.95, and pitch ad-
justing rate r pa =0.7. The search paths are plotted together with the landscape of f(x,y).
From Figure 2, we can see that the pitch adjustment is more intensive in local regions
(two thin strips).
As a further example, we present Michalewicz's bivariate function as follows:

2 2

x 2 y

20 20

(7)

f ( x , y ) = − sin( x ) sin ( ) − sin( y ) sin ( ),

π π

where a global minimum f min ≈ -1.801 at [2.20319, 1.57049] in the domain 0≤x≤π and
0≤y≤π. This global minimum was found by the HS algorithm as shown in Figure 3.
In addition to the above-mentioned two benchmark examples, this topic contains
many successful examples of the HS algorithm in solving various tough optimization
problems, and also provides comparison among the HS algorithm and other ones. Such
a comparison among different types of algorithms is still an area of active research.

Fig. 3. Harmony search for Michalewicz's bivariate function