v2007.09.17 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 65Conversely, given any point y p in R n , the unique hyperplane containingit having normal a is the affine set ∂H (94) where b equals a T y p andwhere a basis for N(a T ) is arranged in Z columnar. Hyperplane dimensionis apparent from the dimensions of Z ; that hyperplane is parallel to thespan 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 . 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 ?2.4.2.0.2 Example. Distance from origin to hyperplane.Given the (shortest) distance ∆∈ R + from the origin to a hyperplanehaving normal vector a , we can find its representation ∂H by droppinga perpendicular. The point thus found is the orthogonal projection of theorigin on ∂H (E.5.0.0.5), equal to a∆/‖a‖ if the origin is known a priorito belong to halfspace H − (Figure 16), or equal to −a∆/‖a‖ if the originbelongs to halfspace H + ; id est, when H − ∋0or when H + ∋0∂H = { y | a T (y − a∆/‖a‖) = 0 } = { y | a T y = ‖a‖∆ } (96)∂H = { y | a T (y + a∆/‖a‖) = 0 } = { y | a T y = −‖a‖∆ } (97)Knowledge of only distance ∆ and normal a thus introduces ambiguity intothe hyperplane representation.2.4.2.1 Matrix variableAny halfspace in R mn may be represented using a matrix variable. Forvariable 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} (98)H + = {Y ∈ R mn | 〈A, Y 〉 ≥ b} = {Y ∈ R mn | 〈A, Y −Y p 〉 ≥ 0} (99)Recall vector inner-product from2.2, 〈A, Y 〉= tr(A T Y ).