Optimization

199Constrained Optimization

Problems

1. For the following optimization problem

Minimize

f(x) = 2x

1

+ x

2

subject to

1 0

1

2

2

+ − ≤x x

check whether the following points are feasible

i. (0, 0)

ii. (1, 2)

iii. (2, 1)

iv. (1, 3)

2. For the following optimization problem

Minimize

f x x( ) ( ) ( )x = − + − +

1

2

2

2

3 4 2

subject to

1 0

1

2

2

+ − ≤x x

check which of the constraints are active at the following points

i. (2, –1)

ii. (1, 2)

iii. (1, 1)

iv. (13/5, 1/5)

3. Solve the following optimization problem using the variable-elimination

method.

Minimize

f(x) = (3x

1

− 2x

2

)

2

+ (x

1

+ 2)

2

subject to

x

1

+ x

2

= 7

200 Optimization: Algorithms and Applications

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