v2009.01.01 - Convex Optimization

202 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.1.3.0.1 Example. Projecting the origin on an affine subset, in 1-norm.
In (1760) we interpret least norm solution to linear system Ax = b as
orthogonal projection of the origin 0 on affine subset A = {x∈ R n |Ax=b}
where A∈ R m×n is fat full-rank. Suppose, instead of the Euclidean metric,
we use taxicab distance to do projection. Then the least 1-norm problem is
stated, for b ∈ R(A)
minimize ‖x‖ 1
x
(460)
subject to Ax = b
Optimal solution can be interpreted as an oblique projection on A simply
because the Euclidean metric is not employed. This problem statement
sometimes returns optimal x ⋆ having minimum cardinality; which can be
explained intuitively with reference to Figure 61: [18]
Projection of the origin, in 1-norm, on affine subset A is equivalent to
maximization (in this case) of the 1-norm ball until it kisses A ; rather, a
kissing point in A achieves the distance in 1-norm from the origin to A . For
the example illustrated (m=1, n=3), it appears that a vertex of the ball will
be first to touch A . 1-norm ball vertices in R 3 represent nontrivial points of
minimum cardinality 1, whereas edges represent cardinality 2, while relative
interiors of facets represent maximum cardinality 3. By reorienting affine
subset A so it were parallel to an edge or facet, it becomes evident as we
expand or contract the ball that a kissing point is not necessarily unique. 3.6
The 1-norm ball in R n has 2 n facets and 2n vertices. 3.7 For n > 0
B 1 = {x∈ R n | ‖x‖ 1 ≤ 1} = conv{±e i ∈ R n , i=1... n} (461)
is a vertex-description of the unit 1-norm ball. Maximization of the 1-norm
ball until it kisses A is equivalent to minimization of the 1-norm ball until it
no longer intersects A . Then projection of the origin on affine subset A is
where
minimize
x∈R n ‖x‖ 1
subject to Ax = b
≡
minimize c
c∈R , x∈R n
subject to x ∈ cB 1
Ax = b
(462)
cB 1 = {[I −I ]a | a T 1=c, a≽0} (463)
3.6 This is unlike the case for the Euclidean ball (1760) where minimum-distance
projection on a convex set is unique (E.9); all kissable faces of the Euclidean ball are
single points (vertices).
3.7 The ∞-norm ball in R n has 2n facets and 2 n vertices.