Engineering Optimization: Theory and Practice, Fourth Edition

7.8 Rosen’s Gradient Projection Method 405

is the vector of Lagrange multipliers associated with Eqs. (7.54) andβis the Lagrange
multiplier associated with Eq. (7.55). The necessary conditions for the minimum are
given by

∂L

∂S

= ∇f (X)+Nλ+ 2 βS= 0 (7.57)

∂L

∂λ

=NTS= 0 (7.58)

∂L

∂β

=STS− 1 = 0 (7.59)

Equation (7.57) gives

S= −

1

2 β

(∇f+Nλ) (7.60)

Substitution of Eq. (7.60) into Eq. (7.58) gives

NTS =−

1

2 β

(NT∇f+NTNλ)= 0 (7.61)

IfSis normalized according to Eq. (7.59),βwill not be zero, and hence Eq. (7.61)
gives

NT∇f+NTNλ= 0 (7.62)

from whichλcan be found as

λ= −(NTN)−^1 NT∇f (7.63)

This equation, when substituted in Eq. (7.60), gives

S= −

1

2 β

(I−N(NTN)−^1 NT) ∇f=−

1

2 β

P∇f (7.64)

where

P=I−N(NTN)−^1 NT (7.65)

is called theprojection matrix. Disregarding the scaling constant 2β, we can say that
the matrixPprojects the vector−∇f (X)onto the intersection of all the hyperplanes
perpendicular to the vectors

∇gj, j=j 1 , j 2 ,... , jp

We assume that the constraintsgj( X)are independent so that the columns of the
matrixNwill be linearly independent, and henceNTN willbe nonsingular and can be
inverted. The vectorScan be normalized [without having to know the value ofβin
Eq. (7.64)] as

S= −

P∇f

||P∇f||

(7.66)