x x x
x
T T T
T
i) ii) iii)
≡ − −
≡
−
−
−
≡ −
≡ −
AOE 5244
Optimization Techniques
HW Set 2
1 2 2 2
1212
1 2 4
11 2 2 2
1 2 1 2 1 2 1
2
1 2
1 2 1 2
1 2 2 2
1
F x , x x x x
, , ,
F x , x x x x x
. , .
F x , x x x / x ,x x
x
x x .
F x , x x
c x , x x x
1. Consider the function
( ) 100( ) + (1 )
Form the expressions for the gradient and the Hessian. Test the optimality of
the following points:
ˆ= (0 0) ˆ= (1 1) ˆ= (1 0)
2. Consider the function
( ) + + (1 + )
Form the expressions for the gradient and the Hessian.
i. Test the optimality of the origin.
ii. Test the optimality of ˆ= ( 6959 1 3479)
3. Consider the (linearly) constrained minimization problem of
min ( ) = + + (1 2)( ) 1 1
1 0
subject to
= 1
Find all stationary points and discuss their optimality.
4. Consider the (nonlinearly) constrained minimization problem of minimiz-
ing the function
( ) 1(1 )
subject to the constraint
( ) ( ) = 0
Find stationary points for the Lagrangian and test the projected Hession (of
the Lagrangian) for definiteness. What points, if any are local minimizers ?
1