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

3.3. PRACTICAL NORM FUNCTIONS, ABSOLUTE VALUE 223
likely to exist vertex solutions to
minimize
x∈R n ‖Ax‖ 1
subject to x ∈ P
optimized over some bounded polyhedron P . 3.10
(502)
3.3.1 k smallest entries
Sum of the k smallest entries of x∈ R n is the optimal objective value from:
for 1≤k ≤n
n∑
n∑
π(x) i = minimize x T y
π(x) i = maximize k t + 1 T z
y∈R n
i=n−k+1 z∈R n , t∈R
subject to 0 ≼ y ≼ 1
≡
subject to x ≽ t1 + z
1 T y = k
z ≼ 0
(503)
i=n−k+1
k∑
i=1
which are dual linear programs, where π(x) 1 = max{x i , i=1... n} where
π is a nonlinear permutation-operator sorting its vector argument into
nonincreasing order. Finding k smallest entries of an n-length vector x is
expressible as an infimum of n!/(k!(n − k)!) linear functions of x . The sum
∑ π(x)i is therefore a concave function of x ; in fact, monotonic (3.7.1.0.1)
in this instance.
3.3.2 k largest entries
Sum of the k largest entries of x∈ R n is the optimal objective value from:
[59, exer.5.19]
π(x) i = maximize x T y
y∈R n
subject to 0 ≼ y ≼ 1
1 T y = k
≡
k∑
π(x) i =
i=1
minimize k t + 1 T z
z∈R n , t∈R
subject to x ≼ t1 + z
z ≽ 0
(504)
3.10 Hint: Suppose, for example, P belongs to an orthant and A were orthogonal. Begin
with A=I and apply level sets of the objective, as in Figure 65 and Figure 68. [ Or rewrite ] x
the problem as a linear program like (492) and (494) but in a composite variable ← y .
t