Description
1. Consider convex functions fi : Rn → R, i = 1, · · · , k. Prove that the set
{x | fi(x) ≤ 0}
is convex.
2. Consider a convex function f : Rn → R. Prove that the set
{(x,t) | f(x) ≤ t}
is convex.
3. (a) If M1,M2 ∈ S2 are positive definite, prove that M1 + M2 is positive definite.
(b) Prove that the set of all n × n positive definite symmetric matrices is convex.
4. Prove that the following set is convex:
!
(x1, x2, x3) ∈ R3 |
”
x1 + x2 x1 − 2×2
x1 − 2×3 x2 + 3×3
#
≽ 0
$
5. Find a necessary and sufficient condition under which the following quadratic function
is convex:
f(x) = %
α1×1 α2×2 · · · αnxn
&
P
‘
(
(
(
)
α1×1
α2×2
.
.
.
αnxn
*
+
+
+
,
.
1