Engineering Optimization: Theory and Practice, Fourth Edition

320 Nonlinear Programming II: Unconstrained Optimization Techniques

and hence

∇Q(X 1 ) −∇Q(X 2 )=A(X 1 −X 2 ) (6.27)

IfSis any vector parallel to the hyperplanes, it must be orthogonal to the gradients

∇Q(X 1 ) nda ∇Q(X 2 ) Thus.

ST∇Q(X 1 )=STAX 1 +STB= 0 (6.28)

ST∇Q(X 2 )=STAX 2 +STB= 0 (6.29)

By subtracting Eq. (6.29) from Eq. (6.28), we obtain

STA(X 1 −X 2 )= 0 (6.30)

HenceSand(X 1 −X 2 ) rea A-conjugate.

The meaning of this theorem is illustrated in a two-dimensional space in Fig. 6.7.

IfX 1 andX 2 are the minima ofQobtained by searching along the directionSfrom two

Figure 6.7 Conjugate directions.