v2009.01.01 - Convex Optimization

70 CHAPTER 2. CONVEX GEOMETRY
where the columns of Z ∈ R n×n−1 constitute a basis for the nullspace
N(a T ) = {x∈ R n | a T x=0} . 2.17
Conversely, given any point y p in R n , the unique hyperplane containing
it having normal a is the affine set ∂H (105) where b equals a T y p and
where a basis for N(a T ) is arranged in Z columnar. Hyperplane dimension
is apparent from the dimensions of Z ; that hyperplane is parallel to the
span of its columns.
2.4.2.0.1 Exercise. Hyperplane scaling.
Given normal y , draw a hyperplane {x∈ R 2 | x T y =1}. Suppose z = 1y . 2
On the same plot, draw the hyperplane {x∈ R 2 | x T z =1}. Now suppose
z = 2y , then draw the last hyperplane again with this new z . What is the
apparent effect of scaling normal y ?
2.4.2.0.2 Example. Distance from origin to hyperplane.
Given the (shortest) distance ∆∈ R + from the origin to a hyperplane
having normal vector a , we can find its representation ∂H by dropping
a perpendicular. The point thus found is the orthogonal projection of the
origin on ∂H (E.5.0.0.5), equal to a∆/‖a‖ if the origin is known a priori
to belong to halfspace H − (Figure 21), or equal to −a∆/‖a‖ if the origin
belongs to halfspace H + ; id est, when H − ∋0
∂H = { y | a T (y − a∆/‖a‖) = 0 } = { y | a T y = ‖a‖∆ } (107)
or when H + ∋0
∂H = { y | a T (y + a∆/‖a‖) = 0 } = { y | a T y = −‖a‖∆ } (108)
Knowledge of only distance ∆ and normal a thus introduces ambiguity into
the hyperplane representation.
2.4.2.1 Matrix variable
Any halfspace in R mn may be represented using a matrix variable. For
variable Y ∈ R m×n , given constants A∈ R m×n and b = 〈A , Y p 〉 ∈ R
H − = {Y ∈ R mn | 〈A, Y 〉 ≤ b} = {Y ∈ R mn | 〈A, Y −Y p 〉 ≤ 0} (109)
H + = {Y ∈ R mn | 〈A, Y 〉 ≥ b} = {Y ∈ R mn | 〈A, Y −Y p 〉 ≥ 0} (110)
2.17 We will later find this expression for y in terms of nullspace of a T (more generally, of
matrix A T (134)) to be a useful device for eliminating affine equality constraints, much as
we did here.