v2010.10.26 - Convex Optimization

76 CHAPTER 2. CONVEX GEOMETRY2.4.2 Hyperplane ∂H representationsEvery hyperplane ∂H is an affine set parallel to an (n −1)-dimensionalsubspace of R n ; it is itself a subspace if and only if it contains the origin.dim∂H = n − 1 (112)so a hyperplane is a point in R , a line in R 2 , a plane in R 3 , and so on. Everyhyperplane can be described as the intersection of complementary halfspaces;[307,19]∂H = H − ∩ H + = {y | a T y ≤ b , a T y ≥ b} = {y | a T y = b} (113)a halfspace-description. Assuming normal a∈ R n to be nonzero, then anyhyperplane in R n can be described as the solution set to vector equationa T y = b (illustrated in Figure 25 and Figure 26 for R 2 );∂H {y | a T y = b} = {y | a T (y−y p ) = 0} = {Zξ+y p | ξ ∈ R n−1 } ⊂ R n (114)All solutions y constituting the hyperplane are offset from the nullspace ofa T by the same constant vector y p ∈ R n that is any particular solution toa T y=b ; id est,y = Zξ + y p (115)where the columns of Z ∈ R n×n−1 constitute a basis for the nullspaceN(a T ) = {x∈ R n | a T x=0} . 2.17Conversely, given any point y p in R n , the unique hyperplane containingit having normal a is the affine set ∂H (114) where b equals a T y p andwhere a basis for N(a T ) is arranged in Z columnar. Hyperplane dimensionis apparent from dimension of Z ; that hyperplane is parallel to the span ofits 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 . 2On the same plot, draw the hyperplane {x∈ R 2 | x T z =1}. Now supposez = 2y , then draw the last hyperplane again with this new z . What is theapparent effect of scaling normal y ?2.17 We will later find this expression for y in terms of nullspace of a T (more generally, ofmatrix A T (144)) to be a useful device for eliminating affine equality constraints, much aswe did here.