v2010.10.26 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 772.4.2.0.2 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 25), or equal to −a∆/‖a‖ if the originbelongs to halfspace H + ; id est, when H − ∋0∂H = { y | a T (y − a∆/‖a‖) = 0 } = { y | a T y = ‖a‖∆ } (116)or when H + ∋0∂H = { y | a T (y + a∆/‖a‖) = 0 } = { y | a T y = −‖a‖∆ } (117)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 〉 ∈ RH − = {Y ∈ R mn | 〈A, Y 〉 ≤ b} = {Y ∈ R mn | 〈A, Y −Y p 〉 ≤ 0} (118)H + = {Y ∈ R mn | 〈A, Y 〉 ≥ b} = {Y ∈ R mn | 〈A, Y −Y p 〉 ≥ 0} (119)Recall vector inner-product from2.2: 〈A, Y 〉= tr(A T Y )= vec(A) T vec(Y ).Hyperplanes in R mn may, of course, also be represented using matrixvariables.∂H = {Y | 〈A, Y 〉 = b} = {Y | 〈A, Y −Y p 〉 = 0} ⊂ R mn (120)Vector a from Figure 25 is normal to the hyperplane illustrated. Likewise,nonzero vectorized matrix A is normal to hyperplane ∂H ;A ⊥ ∂H in R mn (121)2.4.2.2 Vertex-description of hyperplaneAny hyperplane in R n may be described as affine hull of a minimal set ofpoints {x l ∈ R n , l = 1... n} arranged columnar in a matrix X ∈ R n×n : (78)∂H = aff{x l ∈ R n , l = 1... n} ,dim aff{x l ∀l}=n−1= aff X , dim aff X = n−1(122)= x 1 + R{x l − x 1 , l=2... n} , dim R{x l − x 1 , l=2... n}=n−1= x 1 + R(X − x 1 1 T ) , dim R(X − x 1 1 T ) = n−1