Chapter 3 Geometry of convex functions - Meboo Publishing ...

224 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
which are dual linear programs. Finding k largest entries of an n-length
vector x is expressible as a supremum of n!/(k!(n − k)!) linear functions
of x . (Figure 72) The summation is therefore a convex function (and
monotonic in this instance,3.7.1.0.1).
3.3.2.1 k-largest norm
Let Πx be a permutation of entries x i such that their absolute value
becomes arranged in nonincreasing order: |Πx| 1 ≥ |Πx| 2 ≥ · · · ≥ |Πx| n .
Sum of the k largest entries of |x|∈ R n is a norm, by properties of vector
norm (3.3), and is the optimal objective value of a linear program:
k∑ ∑
‖x‖n |Πx| i = k π(|x|) i = minimize k t + 1 T z
k i=1 i=1
z∈R n , t∈R
subject to −t1 − z ≼ x ≼ t1 + z
z ≽ 0
{
= sup a T i x
i∈I
∣ a }
ij ∈ {−1, 0, 1}
carda i = k
= maximize (y 1 − y 2 ) T x
y 1 , y 2 ∈R n
subject to 0 ≼ y 1 ≼ 1
0 ≼ y 2 ≼ 1
where the norm subscript derives from a binomial coefficient
‖x‖n n
= ‖x‖ 1
(y 1 + y 2 ) T 1 = k
( n
k)
, and
(505)
‖x‖n
1
= ‖x‖ ∞
(506)
‖x‖n
k
= ‖π(|x|) 1:k ‖ 1
Sum of k largest absolute entries of an n-length vector x is expressible as
a supremum of 2 k n!/(k!(n − k)!) linear functions of x ; (Figure 72) hence,
this norm is convex (nonmonotonic) in x . [59, exer.6.3e]
minimize ‖x‖n
x∈R n k
subject to x ∈ C
≡
minimize
z∈R n , t∈R , x∈R n
subject to
k t + 1 T z
−t1 − z ≼ x ≼ t1 + z
z ≽ 0
x ∈ C
(507)
3.3.2.1.1 Exercise. Polyhedral epigraph of k-largest norm.
Make those card I = 2 k n!/(k!(n − k)!) linear functions explicit for ‖x‖ 22 and
‖x‖ 21 on R 2 and ‖x‖ 32 on R 3 . Plot ‖x‖ 22 and ‖x‖ 21 in three dimensions.