v2004.06.19 - Convex Optimization

2.3. HALFSPACE, HYPERPLANE 43nonempty interior) 2.14 at each point on its boundary.2.3.2.4.1 Definition. Supporting hyperplane ∂H . The partial boundary∂H of a closed halfspace containing arbitrary set Y is called a supportinghyperplane ∂H to Y when it contains at least one point of Y . [30, §11] Specifically,given normal a ≠0 (belonging to H + by definition), the supportinghyperplane to Y at y p ∈ ∂Y [sic] is∂H − = { y | a T (y − y p ) = 0, y p ∈ Y , a T (z − y p ) ≤ 0 ∀z ∈ Y }= { y | a T y = sup{a T z |z∈Y} } (84)where normal a and set Y reside in opposite halfspaces. (Figure 2.6(a)) Thereal function σ Y (a) ∆ = sup{a T z |z∈Y} is called the support function of Y .An equivalent but non-traditional representation is obtained by reversingthe polarity of normal a ; (1166)∂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} } (85)where normal ã and set Y now reside in H + . (Figure 2.6(b))When the supporting hyperplane contains only a single point of Y , thathyperplane is termed strictly supporting (and termed tangent to Y if thesupporting hyperplane is unique there [30, §18, p.169]).△There is no geometric difference 2.15 between supporting hyperplane ∂H +or ∂H − and an ordinary hyperplane ∂H coincident with them.2.3.2.5 PRINCIPLE 3: Separating hyperplaneThe third most fundamental principle of convex geometry again follows fromthe geometric Hahn-Banach theorem [37, §5.12] [53, §1] that guarantees existenceof a hyperplane separating two nonempty convex sets in R n whoserelative interiors are nonintersecting. Separation intuitively means each setbelongs to a halfspace on an opposing side of the hyperplane. There are twocases of interest:2.14 It is customary to speak of a hyperplane supporting set C but not containing C .[30, p.100]2.15 If vector-normal polarity is unimportant, we may instead represent a supporting hyperplaneby ∂H .