2.5 Multivariable Optimization with Inequality Constraints 97g 2 (X)= 1 −x 1 ≤ 0g 3 (X)= 1 −x 2 ≤ 0g 4 (X)=x 1 − 01 ≤ 0g 5 (X)=x 2 − 01 ≤ 0At the given pointX 1 = { 1 , 7 }T, all the constraints can be seen to be satisfied withg 1
andg 2 being active. The gradients of the objective and active constraint functions at
pointX 1 = { 1 , 7 }Tare given by
∇f=
∂f
∂x 1
∂f
∂x 2
X 1=
2 x 1
2 x 2
X 1=
{
2
14
}
∇g 1 =
∂g 1
∂x 1
∂g 1
∂x 2
X 1=
{
1
2
}
∇g 2 =
∂g 2
∂x 1
∂g 2
∂x 2
X 1=
{
− 1
0
}
For the search directionS= {s 1 , s 2 }T, the usability and feasibility conditions can be
expressed as
(a) Usability condition:ST∇ f≤ 0 or (s 1 s 2 ){
2
14
}
≤0 or 2s 1 + 41 s 2 ≤ 0 (E 1 )(b) Feasibility conditions:ST∇g 1 ≤ 0 or (s 1 s 2 ){
1
2
}
≤ 0 or s 1 + 2 s 2 ≤ 0 (E 2 )ST∇g 2 ≤ 0 or (s 1 s 2 ){
− 1
0
}
≤ 0 or −s 1 ≤ 0 (E 3 )Note:Any two numbers fors 1 ands 2 that satisfy the inequality (E 1 ) ill constitutew
a usable directionS. For example,s 1 = 1 ands 2 = − 1 gives the usable direction
S= { 1 ,− 1 }T. This direction can also be seen to be a feasible direction bec ause it
satisfies the inequalities(E 2 ) nda (E 3 ).