Engineering Optimization: Theory and Practice, Fourth Edition

7.4 Complex Method 385

are found one at a time by the use of random numbers generated in the range

0 to 1, as

xi,j=xi(l)+ri,j(x(u)i −xi(l)), i= 1 , 2 ,... , n, j= 2 , 3 ,... , k (7.8)

wherexi,jis the ith component of the pointXj, andri,jis a random number

lying in the interval (0,1). It is to be noted that the pointsX 2 ,X 3 ,... ,Xk

generated according to Eq. (7.8) satisfy the side constraints, Eqs. (7.7c) but

may not satisfy the constraints given by Eqs. (7.7b).

As soon as a new pointXjis generated (j= 2 , 3 ,... , k), we find whether

it satisfies all the constraints, Eqs. (7.7b). IfXjviolates any of the constraints

stated in Eqs. (7.7b), the trial pointXj is moved halfway toward the centroid

ofthe remaining, already accepted points (where the given initial pointX 1 is

included). The centroidX 0 of already accepted points is given by

X 0 =

1

j− 1

∑j−^1

l= 1

Xl (7.9)

If the trial pointXjso found still violates some of the constraints, Eqs. (7.7b),

the process of moving halfway in toward the centroidX 0 is continued until

afeasible pointXj is found. Ultimately, we will be able to find a feasible

pointXj by this procedure provided that the feasible region is convex. By

proceeding in this way, we will ultimately be able to find the required feasible

pointsX 2 ,X 3 ,... ,Xk.

2 .The objective function is evaluated at each of thekpoints (vertices). If the
vertexXhcorresponds to the largest function value, the process of reflection is
used to find a new pointXras

Xr= ( 1 +α)X 0 −αXh (7.10)

whereα≥1 (to start with) andX 0 is the centroid of all vertices exceptXh:

X 0 =

1

k− 1

∑k

l= 1

l=k

Xl (7.11)

3 .Since the problem is a constrained one, the pointXrhas to be tested for feasi-
bility. If the pointXris feasible andf(Xr) < f (Xh) the point, Xhis replaced
byXr, and we go to step 2. Iff(Xr) ≥f(Xh) a new trial point, Xris found
byreducing the value ofαin Eq. (7.10) by a factor of 2 and is tested for
the satisfaction of the relationf (Xr) < f (Xh) If. f (Xr) ≥f(Xh) the proce-,
dure of finding a new pointXrwith a reduced value ofαis repeated again.
This procedure is repeated, if necessary, until the value ofαbecomes smaller
than a prescribed small quantityε, say, 10−^6. If an improved pointXr, with
f(Xr) < f (Xh) cannot be obtained even with that small value of, α, the point
Xris discarded and the entire procedure of reflection is restarted by using the
pointXp(which has the second-highest function value) instead ofXh.