414 Nonlinear Programming III: Constrained Optimization Techniques
Consider the first variations of the objective and constraint functions:df (X)=∑n−li= 1∂f
∂yidyi+∑m+li= 1∂f
∂zidzi= ∇TYf d Y+∇TZf dZ (7.94)dgi(X)=∑n−lj= 1∂gi
∂yjdyj+m∑+lj= 1∂gi
∂zjdzjor
dg=[C]dY+[D]dZ (7.95)where∇Yf=
∂f
∂y 1
∂f
∂y 2
..
.
∂f
∂yn−l
(7.96)
∇Zf=
∂f
∂z 1
∂f
∂z 2
..
.
∂f
∂zm+l
(7.97)
[C]=
∂g 1
∂y 1· · ·
∂g 1
∂yn−l
..
...
.
∂gm+l
∂y 1· · ·
∂gm+l
∂yn−l
(7.98)
[D]=
∂g 1
∂z 1· · ·
∂g 1
∂zm+l
..
...
.
∂gm+l
∂z 1· · ·
∂gm+l
∂zm+l