6.7 Simplex Method 331
Whenever such situation is encountered, we reject the vertex corresponding to the
second worst value instead of the vertex corresponding to the worst function value.
This method, in general, leads the process to continue toward the region of the desired
minimum. However, the final simplex may again straddle the minimum, or it may lie
within a distance of the order of its own size from the minimum. In such cases it may
not be possible to obtain a new simplex with vertices closer to the minimum compared
to those of the previous simplex, and the pattern may lead to a cyclic process, as shown
in Fig. 6.13. In this example the successive simplexes formed from the simplex 123
are 234, 245, 456, 467, 478, 348, 234, 245,.. .,†which can be seen to be forming
acyclic process. Whenever this type of cycling is observed, one can take the vertex
that is occurring in every simplex (point 4 in Fig. 6.13) as the best approximation to
the optimum point. If more accuracy is desired, the simplex has to be contracted or
reduced in size, as indicated later.
6.7.2 Expansion
If a reflection process gives a pointXrfor which f(Xr) < f (Xl) (i.e., if the reflection,
produces a new minimum), one can generally expect to decrease the function value
further by moving along the direction pointing fromX 0 toXr. Hence we expandXr
Figure 6.13 Reflection process leading to a cyclic process.
†Simplexes 456, 467, and 234 are formed by reflecting the second-worst point to avoid the difficulty
mentioned earlier.
