v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
74 CHAPTER 2. CONVEX GEOMETRY tradition (a) Y y p a H + H − ∂H − nontraditional (b) Y y p ã H − H + ∂H + Figure 25: (a) Hyperplane ∂H − (119) supporting closed set Y ∈ R 2 . Vector a is inward-normal to hyperplane with respect to halfspace H + , but outward-normal with respect to set Y . A supporting hyperplane can be considered the limit of an increasing sequence in the normal-direction like that in Figure 24. (b) Hyperplane ∂H + nontraditionally supporting Y . Vector ã is inward-normal to hyperplane now with respect to both halfspace H + and set Y . Tradition [173] [266] recognizes only positive normal polarity in support function σ Y as in (120); id est, normal a , figure (a). But both interpretations of supporting hyperplane are useful.
2.4. HALFSPACE, HYPERPLANE 75 2.4.2.6 PRINCIPLE 2: Supporting hyperplane The second most fundamental principle of convex geometry also follows from the geometric Hahn-Banach theorem [215,5.12] [17,1] that guarantees existence of at least one hyperplane in R n supporting a convex set (having nonempty interior) 2.18 at each point on its boundary. The partial boundary ∂H of a halfspace that contains arbitrary set Y is called a supporting hyperplane ∂H to Y when the hyperplane contains at least one point of Y . [266,11] 2.4.2.6.1 Definition. Supporting hyperplane ∂H . Assuming set Y and some normal a≠0 reside in opposite halfspaces 2.19 (Figure 25(a)), then a hyperplane supporting Y at y p ∈ ∂Y is ∂H − = { y | a T (y − y p ) = 0, y p ∈ Y , a T (z − y p ) ≤ 0 ∀z∈Y } (119) Given normal a , instead, the supporting hyperplane is where real function ∂H − = { y | a T y = sup{a T z |z∈Y} } (120) σ Y (a) ∆ = sup{a T z |z∈Y} (491) is called the support function for Y . An equivalent but nontraditional representation 2.20 for a supporting hyperplane is obtained by reversing polarity of normal a ; (1560) ∂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} } (121) where normal ã and set Y both now reside in H + . (Figure 25(b)) When a supporting hyperplane contains only a single point of Y , that hyperplane is termed strictly supporting. 2.21 △ 2.18 It is conventional to speak of a hyperplane supporting set C but not containing C ; called nontrivial support. [266, p.100] Hyperplanes in support of lower-dimensional bodies are admitted. 2.19 Normal a belongs to H + by definition. 2.20 useful for constructing the dual cone; e.g., Figure 48(b). Tradition would instead have us 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; [266,18, p.169] a definition we do not adopt because our only criterion for tangency is intersection exclusively with a relative boundary. Hiriart-Urruty & Lemaréchal [173, p.44] (confer [266, p.100]) do not demand any tangency of a supporting hyperplane.
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74 CHAPTER 2. CONVEX GEOMETRY<br />
tradition<br />
(a)<br />
Y<br />
y p<br />
a<br />
H +<br />
H −<br />
∂H −<br />
nontraditional<br />
(b)<br />
Y<br />
y p<br />
ã<br />
H −<br />
H +<br />
∂H +<br />
Figure 25: (a) Hyperplane ∂H − (119) supporting closed set Y ∈ R 2 .<br />
Vector a is inward-normal to hyperplane with respect to halfspace H + ,<br />
but outward-normal with respect to set Y . A supporting hyperplane can<br />
be considered the limit of an increasing sequence in the normal-direction like<br />
that in Figure 24. (b) Hyperplane ∂H + nontraditionally supporting Y .<br />
Vector ã is inward-normal to hyperplane now with respect to both<br />
halfspace H + and set Y . Tradition [173] [266] recognizes only positive<br />
normal polarity in support function σ Y as in (120); id est, normal a ,<br />
figure (a). But both interpretations of supporting hyperplane are useful.
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