v2010.10.26 - Convex Optimization

82 CHAPTER 2. CONVEX GEOMETRY2.4.2.6.1 Definition. Supporting hyperplane ∂H .Assuming set Y and some normal a≠0 reside in opposite halfspaces 2.19(Figure 29a), then a hyperplane supporting Y at point y p ∈ ∂Y is described∂H − = { y | a T (y − y p ) = 0, y p ∈ Y , a T (z − y p ) ≤ 0 ∀z∈Y } (128)Given only normal a , the hyperplane supporting Y is equivalently describedwhere real function∂H − = { y | a T y = sup{a T z |z∈Y} } (129)σ Y (a) = sup{a T z |z∈Y} (553)is called the support function for Y .Another equivalent but nontraditional representation 2.20 for a supportinghyperplane is obtained by reversing polarity of normal a ; (1678)∂H + = { y | ã T (y − y p ) = 0, y p ∈ Y , ã T (z − y p ) ≥ 0 ∀z∈Y }= { y | ã T y = − inf{ã T z |z∈Y} = sup{−ã T z |z∈Y} } (130)where normal ã and set Y both now reside in H + (Figure 29b).When a supporting hyperplane contains only a single point of Y , thathyperplane is termed strictly supporting. 2.21△A full-dimensional set that has a supporting hyperplane at every pointon its boundary, conversely, is convex. A convex set C ⊂ R n , for example,can be expressed as the intersection of all halfspaces partially bounded byhyperplanes supporting it; videlicet, [250, p.135]C = ⋂a∈R n {y | a T y ≤ σ C (a) } (131)by the halfspaces theorem (2.4.1.1.1).There is no geometric difference between supporting hyperplane ∂H + or∂H − or ∂H and 2.22 an ordinary hyperplane ∂H coincident with them.2.19 Normal a belongs to H + by definition.2.20 useful for constructing the dual cone; e.g., Figure 55b. Tradition would instead haveus construct the polar cone; which is, the negative dual cone.2.21 Rockafellar terms a strictly supporting hyperplane tangent to Y if it is unique there;[307,18, p.169] a definition we do not adopt because our only criterion for tangency isintersection exclusively with a relative boundary. Hiriart-Urruty & Lemaréchal [199, p.44](confer [307, p.100]) do not demand any tangency of a supporting hyperplane.2.22 If vector-normal polarity is unimportant, we may instead signify a supportinghyperplane by ∂H .