Engineering Optimization: Theory and Practice, Fourth Edition

6.7 Simplex Method 329

Figure 6.10 Reflection.

by rejecting the vertex corresponding to the highest function value. Since the direction
of movement of the simplex is always away from the worst result, we will be moving
in a favorable direction. If the objective function does not have steep valleys, repetitive
application of the reflection process leads to a zigzag path in the general direction of
the minimum as shown in Fig. 6.11. Mathematically, the reflected pointXris given
by

Xr= ( 1 +α)X 0 −αXh (6.48)

whereXhis the vertex corresponding to the maximum function value:

f(Xh) = max

i = 1 ton+ 1

f (Xi), (6.49)

Figure 6.11 Progress of the reflection process.