Harmony Search Methods for Multi-modal and Constrained Optimization - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 48

Each nonlinear function has only one global optimum (minimum) but several local
optima. The goal of the optimization algorithms is to find not only the global opti-
mum but also as many local optima as possible. The optimization results of the regu-
lar HS and modified HS methods are illustrated in Figs. 5 and 6. HMS = 100, HMCR
= 0.75, and PAR = 0.6 are used in both HS techniques. In our modified HS method,
V and M for f 1 ( x , y ) and f 2 ( x , y ) are:

V

for f 1 ( x , y ) , V = 0 . 15 , and M = 2

V

for f 2 ( x , y ) , V = 0 . 075 , and M = 3

V

It is clearly visible that the regular HS method can only find the global optimum,
while the modified HS method is capable of locating most of the local optima in addi-
tion to the global optimum. However, we emphasize that V and M can affect the
multi-modal optimization performance of our modified HS method. In case of a fixed
M , if V is too small, the behaviors of the normal HS and modified HS methods are
similar. On the other hand, the performance of the modified HS method can signifi-
cantly deteriorate in case of a too large V . Unfortunately, like other HS parameters,
how to choose the best V and M is still an unsolved problem, although some adap-
tation strategies can be the potential solutions.

5.2 Constrained Optimization Problems

Two numerical examples are used here to demonstrate the capability of the second
modified HS method in handling constrained optimization problems. The first exam-
ple is a function optimization problem with two variables, x and

x , and two con-

g 1 x G and g 2 x G , as follows: straints, ( ) ( )

x G

Minimize f( ) = ( x

2
1

+ x − 11 )

2

+ ( x + x

2
2

− 7 )

2

,

2 1

subject to

x G g ( ) = ( x − 0 . 05 )

2

+ ( x − 2 .

2

− 4 . 84 ≤ 0 ,

1 1 2

x G g ( ) = 4 84 − x

2
1

− ( x − 2 . 5

2

≤ 0 ,

2 2

0 ≤ x ≤ 6 , 0 ≤ x ≤ 6 .

1 2

The modified HS method successfully found the optimal solution without violating
any constraint while a gradient-based method (BFGS) failed to find feasible solution.
The second example is the optimal design of the welded beam. The goal here is to
minimize the fabricating cost of the welded beam subject to the constraints on the
shear stress τ , bending stress on the beam σ , buckling load on the bar P , end de-

flection of the beam δ , and side constraints. The four design variables ( h , l , t , and

b ) are denoted as x ,

x , x , and x , as shown in Fig. 7. The details of the welded

beam design problem are: