Optimization

426 Appendix C

9. Global minimum x* = (−0.656, −0.656) with f(x*) = −2.661

10. Global minimum x* = (1.139, 0.8996) with f(x*) = 1.9522

11. Global minimum x* = (1, 3) with f(x*) = 0

12. Global minimum x* = (0, 0) with f(x*) = 0

Chapter 6

1.

i. Infeasible

ii. Feasible

iii. Infeasible

iv. Feasible

2. Only ii is active.

3. Substitute the value of x

2

= 7 − x

1

in the objective function

f(x) = (5x

1

− 14)

2

+ (x

1

+ 2)

2

Taking the rst derivative as zero gives,

x

1

34

13

=

Therefore,

x

2

57

13

=

and

f ( *)x =

288

13

4. Writing the Lagrangian as

L(x,λ) = (3x

1

− 2x

2

)

2

+ (x

1

+ 2)

2

+ λ(x

1

+ x

2

− 7)

The KKT conditions are given by the equations ...