Numerical Methods and Optimization

366 Numerical Methods and Optimization: An Introduction

The Lagrangian is

L(x, λ, μ)=x

2

− x

1

+ λ(x

2

1

+ x

2

2

− 4) + μ((x

1

+1)

2

+ x

2

2

− 4).

Using the KKT conditions we have the system

−1+2λx

1

+2μ(x

1

+1) = 0

1+2λx

2

+2μx

2

=0

μ

$

(x

1

+1)

2

+ x

2

2

− 4

%

=0

x

2

1

+ x

2

2

=4

(x

1

+1)

2

+ x

2

2

≤ 4

μ ≥ 0.

To solve this system, we consider two cases: μ =0and (x

1

+1)

2

+ x

2

2

−4=0.

1. In the first case, the KKT system reduces to

−1+2λx

1

=0

1+2λx

2

=0

x

2

1

+ x

2

2

=4

(x

1

+1)

2

+ x

2

2

≤ 4.

From the first two equations, noting that λ cannot be zero, we obtain

x

1

= −x

2

, and considering the third equation we have x

1

= −x

2

=

±

√

2. Taking into account the inequality constraint, we obtain a unique

solution to the above system,

x

(1)

1

=