7.10 Sequential Quadratic Programming 425
subject to
gj+ ∇gjT X≤ 0 , j= 1 , 2 ,... , m
hk+ ∇hTk X= 0 , k= 1 , 2 ,... , p (7.136)
with the Lagrange function given by
L ̃=f (X)+
∑m
j= 1
λjgj(X)+
∑p
k= 1
λm+khk(X) (7.137)
Since the minimum of the augmented Lagrange function is involved, the sequential
quadratic programming method is also known as theprojected Lagrangian method.
7.10.2 Solution Procedure
As in the case of Newton’s method of unconstrained minimization, the solution vector
Xin Eq. (7.136) is treated as the search direction,S, and the quadratic programming
subproblem (in terms of the design vectorS) is restated as:
FindSwhich minimizesQ(S)= ∇f (X)TS+^12 ST[H]S
subjectto
βjgj (X)+∇gj(X)TS ≤ 0 , j= 1 , 2 ,... , m
βhk( X)+∇hk(X)TS = 0 , k= 1 , 2 ,... , p (7.138)
where [H] is a positive definite matrix that is taken initially as the identity matrix
and is updated in subsequent iterations so as to converge to the Hessian matrix of the
Lagrange function of Eq. (7.137), andβjandβare constants used to ensure that the
linearized constraints do not cut off the feasible space completely. Typical values of
these constants are given by
β≈ 0. 9 ; βj=
{
1 ifgj(X)≤ 0
β ifgj(X)≥ 0
(7.139)
The subproblem of Eq. (7.138) is a quadratic programming problem and hence the
method described in Section 4.8 can be used for its solution. Alternatively, the problem
can be solved by any of the methods described in this chapter since the gradients of the
function involved can be evaluated easily. Since the Lagrange multipliers associated
with the solution of the problem, Eq. (7.138), are needed, they can be evaluated using
Eq. (7.263). Once the search direction,S, is found by solving the problem in Eq. (7.138),
the design vector is updated as
Xj+ 1 =Xj+α∗S (7.140)
whereα∗is the optimal step length along the directionSfound by minimizing the
function (using an exterior penalty function approach):
φ=f (X)+
∑m
j= 1
λj( ax[0m , gj(X)])+
∑p
k= 1
λm+k|hk(X)| (7.141)
