v2007.09.13 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 63Intersection of multiple halfspaces in R n may be represented using amatrix constant A ;⋂H i−= {y | A T y ≼ b} = {y | A T (y − y p ) ≼ 0} (90)i⋂H i+= {y | A T y ≽ b} = {y | A T (y − y p ) ≽ 0} (91)iwhere b is now a vector, and the i th column of A is normal to a hyperplane∂H i partially bounding H i . By the halfspaces theorem, intersections likethis can describe interesting convex Euclidean bodies such as polyhedra andcones, giving rise to the term halfspace-description.2.4.2 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 (92)so a hyperplane is a point in R , a line in R 2 , a plane in R 3 , and so on.Every hyperplane can be described as the intersection of complementaryhalfspaces; [228,19]∂H = H − ∩ H + = {y | a T y ≤ b , a T y ≥ b} = {y | a T y = b} (93)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 16 and Figure 17 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 (94)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 (95)where the columns of Z ∈ R n×n−1 constitute a basis for the nullspaceN(a T ) = {x∈ R n | a T x=0} . 2.162.16 We will later find this expression for y in terms of nullspace of a T (more generally, ofmatrix A T (122)) to be a useful device for eliminating affine equality constraints, much aswe did here.