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

222 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
Then (498) is equivalent to
minimize c
c∈R , x∈R n , a∈R 2n
subject to x = [I −I ]a
a T 1 = c
a ≽ 0
Ax = b
≡
minimize
a∈R 2n ‖a‖ 1
subject to [A −A ]a = b
a ≽ 0
(500)
where x ⋆ = [I −I ]a ⋆ . (confer (495)) Significance of this result:
(confer p.375) Any vector 1-norm minimization problem may have its
variable replaced with a nonnegative variable of twice the length.
All other things being equal, nonnegative variables are easier to solve for
sparse solutions. (Figure 69, Figure 70, Figure 97) The compressed sensing
problem becomes easier to interpret; e.g., for A∈ R m×n
minimize ‖x‖ 1
x
subject to Ax = b
x ≽ 0
≡
minimize 1 T x
x
subject to Ax = b
x ≽ 0
(501)
movement of a hyperplane (Figure 26, Figure 30) over a bounded polyhedron
always has a vertex solution [92, p.22]. Or vector b might lie on the relative
boundary of a pointed polyhedral cone K = {Ax | x ≽ 0}. In the latter
case, we find practical application of the smallest face F containing b from
2.13.4.3 to remove all columns of matrix A not belonging to F ; because
those columns correspond to 0-entries in vector x .
3.3.0.0.2 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.
What about minimizing other functions? Given some nonsingular
matrix A , geometrically describe three circumstances under which there are