v2009.01.01 - Convex Optimization

k∑
i=1
3.1. CONVEX FUNCTION 205
Sum of the k largest entries of x∈ R n is the optimal objective value from:
[53, exer.5.19]
π(x) i = maximize x T y
y∈R n
subject to 0 ≼ y ≼ 1
1 T y = k
or
k∑
π(x) i =
i=1
minimize k t + 1 T z
z∈R n , t∈R
subject to x ≼ t1 + z
z ≽ 0
(468)
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 . The summation is therefore a convex function (and monotonic in this
instance,3.1.8.1.1).
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 [197, p.59] [134, p.52], and is the optimal objective value of a linear
program:
‖x‖n
k ∆ = k ∑
i=1
|Πx| i = minimize k t + 1 T z
z∈R n , t∈R
subject to −t1 − z ≼ x ≼ t1 + z
z ≽ 0
where the norm subscript derives from a binomial coefficient
( n
k)
, and
(469)
‖x‖n n ∆ = ‖x‖ 1
‖x‖n
1 ∆ = ‖x‖ ∞
(470)
Finding 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 ; hence, this norm is
convex (nonmonotonic) in x . [53, exer.6.3(e)]
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
(471)