Homework2_2011 - f x 1 ∥ x 3 − x 1 ∥ ≥ f x 2 − f...

ESE 415 Optimization
Assignment 2
Due: February 16, 2011
1. Consider f ( x ) = 2 x
2
1
+ x
1
x
2
+ x
2
2
+ x
2
x
3
+ x
2
3
− 6 x
1
− 7 x
2
− 8 x
3
+ 9.
(a) Compute the gradient ∇ f ( x ) and Hessian H ( x ) of f .
(b) Evaluate ∇ f ( x ) and H ( x ) at x = 0 =


0
0
0


.
(c) Obtain the second order approximation of f at x = 0 , i.e., the Taylor series up
to the 2nd order
g ( x ) = f ( x
*
) + ( x − x
*
)
T
∇ f ( x
*
) +
1
2
( x − x
*
)
T
∇
2
f ( x
*
)( x − x
*
) ,
where x = 0 . Is g ( x ) identical to f ( x )?
2. Show the following convexity results.
(a) If f : R
n
→ R is a convex function and x = Ay + b , where A ∈ R
n × n
and b ∈ R
n
,
then g ( y ) = f ( Ay + b ) is convex.
(b) Let C ⊂ R n be a convex set and let f : C → R be twice continuously differentiable
over C . If C = R
n
and f is convex, then H ( x ) ≽ 0, ∀ x ∈ C (Hint: show this by
contradiction and use the continuity of ∇ 2 f ).
(c) Show that if f
1
and f
2
are convex functions then their pointwise maximum f ,
defined by
f ( x ) = max { f
1
( x ) , f
2
( x ) } ,
with dom f = dom f
1
∩ dom f
2
, is also convex, where dom f denotes the domain
of f .
(d) Let x
1
, x
2
, x
3
∈ R n such that x
2
∈ ( x
1
, x
3
). Show that if f is convex in R n then
f ( x
3
) −