v2009.01.01 - Convex Optimization

204 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.1.3.0.2 Exercise. Combinatorial optimization.
A device commonly employed to relax combinatorial problems is to arrange
desirable solutions at vertices of bounded polyhedra; e.g., the permutation
matrices of dimension n , which are factorial in number, are the extreme
points of a polyhedron in the nonnegative orthant described by an
intersection of 2n hyperplanes (2.3.2.0.4). Minimizing a linear objective
function over a bounded polyhedron is a convex problem (a linear program)
that always has an optimal solution residing at a vertex. [82]
What about minimizing other functions? Given some nonsingular
matrix A , geometrically describe three circumstances under which there are
likely to exist vertex solutions to
minimize
x∈R n ‖Ax‖ 1
subject to x ∈ P
optimized over some bounded polyhedron P . 3.8
(466)
3.1.3.1 k smallest/largest entries
Sum of the k smallest entries of x∈ R n is the optimal objective value from:
for 1≤k ≤n
n∑
π(x) i =
i=n−k+1
minimize x T y
y∈R n
subject to 0 ≼ y ≼ 1
1 T y = k
or
n∑
π(x) i = maximize
i=n−k+1
z∈R n , t∈R
k t + 1 T z
subject to x ≽ t1 + z
z ≼ 0
(467)
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 (3.1.8.1.1)
in this instance.
3.8 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 59 and Figure 61. [ Or rewrite ] x
the problem as a linear program like (456) and (458) but in a composite variable ← y .
t