Engineering Optimization: Theory and Practice, Fourth Edition

332 Nonlinear Programming II: Unconstrained Optimization Techniques

toXeusing the relation

Xe=γXr+ ( 1 −γ)X 0 (6.53)

whereγis called theexpansion coefficient, defined as

γ=

distance betweenXeandX 0

distance betweenXrandX 0

> 1

Iff (Xe) < f (Xl) we replace the point, XhbyXeand restart the process of reflec-

tion. On the other hand, iff (Xe) >f(Xl) it means that the expansion process is not,

successful and hence we replace pointXhbyXrand start the reflection process again.

6.7.3 Contraction

If the reflection process gives a pointXrfor whichf(Xr) >f(Xi) or allf iexcept

i=h, andf (Xr) < f (Xh) we replace point, XhbyXr. Thus the newXhwill beXr.

In this case we contract the simplex as follows:

Xc=βXh+ ( 1 −β)X 0 (6.54)

whereβis called thecontraction coefficient(0≤β≤1) and is defined as

β=

distance betweenXeandX 0

distance betweenXhandX 0

If f(Xr) >f(Xh) we still use Eq. (6.54) without changing the previous point, Xh. If

the contraction process produces a pointXcfor which f(Xc) < min[f (Xh), f (Xr) , we]

replace the pointXhinX 1 ,X 2 ,... ,Xn+ 1 byXcand proceed with the reflection process

again.On the other hand, iff (Xc) ≥min[f (Xh), f (Xr) the contraction process will],

be a failure, and in this case we replace allXiby(Xi+Xl)/ and restart the reflection 2

process.

The method is assumed to have converged whenever the standard deviation of the

function at then+1 vertices of the current simplex is smaller than some prescribed

small quantityε, that is,

Q=

{n+ 1

∑

i= 1

[ (f Xi) −f(X 0 )]^2

n+ 1

} 1 / 2

≤ε (6.55)

Example 6.7 Minimize f (x 1 , x 2 )=x 1 −x 2 + 2 x 12 + 2 x 1 x 2 +x 22. Take the points

defining the initial simplex as

X 1 =

{

4. 0

4. 0

}

, X 2 =

{

5. 0

4. 0

}

, and X 3 =

{

4. 0

5. 0

}

andα= 1. 0 , β= 0 .5, andγ= 2 .0. For convergence, take the value ofεas 0.2.