v2007.09.13 - Convex Optimization
v2007.09.13 - Convex Optimization v2007.09.13 - Convex Optimization
68 CHAPTER 2. CONVEX GEOMETRYCH −H +0 > κ 3 > κ 2 > κ 1{z ∈ R 2 | a T z = κ 1 }a{z ∈ R 2 | a T z = κ 2 }{z ∈ R 2 | a T z = κ 3 }Figure 19: Each shaded line segment {z ∈ C | a T z = κ i } belonging to setC ⊂ R 2 shows intersection with hyperplane parametrized by scalar κ i ; eachshows a (linear) contour in vector z of equal inner product with normalvector a . Cartesian axes drawn for reference. (confer Figure 55)2.4.2.5 affine mapsAffine transformations preserve affine hulls. Given any affine mapping T ofvector spaces and some arbitrary set C [228, p.8]aff(T C) = T(aff C) (107)2.4.2.6 PRINCIPLE 2: Supporting hyperplaneThe second most fundamental principle of convex geometry also follows fromthe geometric Hahn-Banach theorem [181,5.12] [16,1] that guaranteesexistence of at least one hyperplane in R n supporting a convex set (having
2.4. HALFSPACE, HYPERPLANE 69tradition(a)Yy paH +H −∂H −nontraditional(b)Yy pãH −H +∂H +Figure 20: (a) Hyperplane ∂H − (108) 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 canbe considered the limit of an increasing sequence in the normal-direction likethat in Figure 19. (b) Hyperplane ∂H + nontraditionally supporting Y .Vector ã is inward-normal to hyperplane now with respect to bothhalfspace H + and set Y . Tradition [147] [228] recognizes only positivenormal polarity in support function σ Y as in (108); id est, normal a ,figure (a). But both interpretations of supporting hyperplane are useful.
- Page 17 and 18: LIST OF FIGURES 17E Projection 5791
- Page 19 and 20: Chapter 1OverviewConvex Optimizatio
- Page 21 and 22: ˇx 4ˇx 3ˇx 2Figure 2: Applicatio
- Page 23 and 24: 23Figure 4: This coarsely discretiz
- Page 25 and 26: ases (biorthogonal expansion). We e
- Page 27 and 28: 27Figure 7: These bees construct a
- Page 29 and 30: its membership to the EDM cone. The
- Page 31 and 32: 31appendicesProvided so as to be mo
- Page 33 and 34: Chapter 2Convex geometryConvexity h
- Page 35 and 36: 2.1. CONVEX SET 35Figure 9: A slab
- Page 37 and 38: 2.1. CONVEX SET 372.1.6 empty set v
- Page 39 and 40: 2.1. CONVEX SET 392.1.7.1 Line inte
- Page 41 and 42: 2.1. CONVEX SET 41(a)R 2(b)R 3(c)(d
- Page 43 and 44: 2.1. CONVEX SET 43This theorem in c
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- Page 53 and 54: 2.3. HULLS 53Figure 12: Convex hull
- Page 55 and 56: 2.3. HULLS 55Aaffine hull (drawn tr
- Page 57 and 58: 2.3. HULLS 57The union of relative
- Page 59 and 60: 2.4. HALFSPACE, HYPERPLANE 59of dim
- Page 61 and 62: 2.4. HALFSPACE, HYPERPLANE 61H +ay
- Page 63 and 64: 2.4. HALFSPACE, HYPERPLANE 63Inters
- Page 65 and 66: 2.4. HALFSPACE, HYPERPLANE 65Conver
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- Page 73 and 74: 2.5. SUBSPACE REPRESENTATIONS 732.5
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- Page 77 and 78: 2.6. EXTREME, EXPOSED 77In other wo
- Page 79 and 80: 2.6. EXTREME, EXPOSED 792.6.1 Expos
- Page 81 and 82: 2.7. CONES 812.6.1.3.1 Definition.
- Page 83 and 84: 2.7. CONES 830Figure 24: Boundary o
- Page 85 and 86: 2.7. CONES 852.7.2 Convex coneWe ca
- Page 87 and 88: 2.7. CONES 87Thus the simplest and
- Page 89 and 90: 2.7. CONES 89nomenclature generaliz
- Page 91 and 92: 2.8. CONE BOUNDARY 91Proper cone {0
- Page 93 and 94: 2.8. CONE BOUNDARY 93the same extre
- Page 96 and 97: 96 CHAPTER 2. CONVEX GEOMETRYBCADFi
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2.4. HALFSPACE, HYPERPLANE 69tradition(a)Yy paH +H −∂H −nontraditional(b)Yy pãH −H +∂H +Figure 20: (a) Hyperplane ∂H − (108) 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 canbe considered the limit of an increasing sequence in the normal-direction likethat in Figure 19. (b) Hyperplane ∂H + nontraditionally supporting Y .Vector ã is inward-normal to hyperplane now with respect to bothhalfspace H + and set Y . Tradition [147] [228] recognizes only positivenormal polarity in support function σ Y as in (108); id est, normal a ,figure (a). But both interpretations of supporting hyperplane are useful.