100 Classical Optimization Techniques
Figure 2.9 Feasible region and contours of the objective function.It is clear that∇g 1 (X∗) nda ∇g 2 (X∗) re not linearly independent. Hence the constrainta
qualification is not satisfied at the optimum point. Noting that∇f (X∗)={
2 (x 1 − 1 )
2 x 2}
( 0 , 0 )=
{
− 2
0
}
the Kuhn–Tucker conditions can be written, using Eqs. (2.73) and (2.74), as− 2 +λ 1 ( 0 )+λ 2 ( 0 )= 0 (E 4 )0 +λ 1 (− 2 )+λ 2 ( 2 )= 0 (E 5 )
λ 1 > 0 (E 6 )
λ 2 > 0 (E 7 )Since Eq.(E 4 ) s not satisfied and Eq.i (E 5 ) an be satisfied for negative values ofc
λ 1 =λ 2 also, the Kuhn–Tucker conditions are not satisfied at the optimum point.