Engineering Optimization: Theory and Practice, Fourth Edition

2.5 Multivariable Optimization with Inequality Constraints 103

This solution can be seen to satisfy Eqs.(E 10 ) ot (E 12 ) ut violate Eqs.b (E 7 )

and(E 9 ).

3 .λ 2 = 0 ,λ 3 =. Equations 0 (E 13 ) iveg

x 1 = − 20 , x 2 = − 10 , x 3 = 0

This solution satisfies Eqs.(E 10 ) ot (E 12 ) ut violates the constraints, Eqs.b (E 7 )

to(E 9 ).

4 .− 130 −λ 2 −λ 3 = 0 ,− 180 −λ 2 −^32 λ 3 =. The solution of these equations 0

and Eqs.(E 13 ) ieldsy

λ 2 = − 30 , λ 3 = − 100 , x 1 = 54 , x 2 = 55 , x 3 = 05

This solution satisfies Eqs.(E 10 ) ot (E 12 ) ut violates the constraint, Eq.b (E 7 ).

Case 2:x 1 = 05.

In this case, Eqs.(E 1 ) ot (E 3 ) iveg

λ 3 = − 2 x 3

λ 2 = − 20 − 2 x 2 −λ 3 = − 20 − 2 x 2 + 2 x 3

λ 1 = − 40 − 2 x 1 −λ 2 −λ 3 = − 120 + 2 x 2

(E 15 )

Substitution of Eqs.(E 15 ) n Eqs.i (E 5 ) nda (E 6 ) eads tol

(− 20 − 2 x 2 + 2 x 3 )(x 1 +x 2 − 001 )= 0

(− 2 x 3 )(x 1 +x 2 +x 3 − 501 )= 0 (E 16 )

Onceagain, it can be seen that there are four possible solutions to Eqs.(E 16 ) as,
indicated below:

1.− 20 − 2 x 2 + 2 x 3 = 0 ,x 1 +x 2 +x 3 − 501 =0: The solution of these

equations yields

x 1 = 05 , x 2 = 54 , x 3 = 55

This solution can be seen to violate Eq.(E 8 ).

2 .− 20 − 2 x 2 + 2 x 3 = 0 ,− 2 x 3 = : These equations lead to the solution 0

x 1 = 05 , x 2 = − 10 , x 3 = 0

This solution can be seen to violate Eqs.(E 8 ) nda (E 9 ).

3 .x 1 +x 2 − 001 = 0 , − 2 x 3 = : These equations give 0

x 1 = 05 , x 2 = 05 , x 3 = 0

This solution violates the constraint Eq.(E 9 ).