v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
80 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 28: (confer Figure 73) Each linear contour, of equal inner product invector z with normal a , represents i th hyperplane in R 2 parametrized byscalar κ i . Inner product κ i increases in direction of normal a . i th linesegment {z ∈ C | a T z = κ i } in convex set C ⊂ R 2 represents intersection withhyperplane. (Cartesian axes drawn for reference.)
2.4. HALFSPACE, HYPERPLANE 81tradition(a)Yy paH +H −∂H −nontraditional(b)Yy pãH −H +∂H +Figure 29: (a) Hyperplane ∂H − (128) 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 28. (b) Hyperplane ∂H + nontraditionally supporting Y .Vector ã is inward-normal to hyperplane now with respect to bothhalfspace H + and set Y . Tradition [199] [307] recognizes only positivenormal polarity in support function σ Y as in (129); id est, normal a ,figure (a). But both interpretations of supporting hyperplane are useful.
- Page 29 and 30: 29cardinality Boolean solution to a
- Page 31 and 32: 31Figure 8: Robotic vehicles in con
- Page 33 and 34: an elaborate exposition offering in
- Page 35 and 36: Chapter 2Convex geometryConvexity h
- Page 37 and 38: 2.1. CONVEX SET 372.1.2 linear inde
- Page 39 and 40: 2.1. CONVEX SET 392.1.6 empty set v
- Page 41 and 42: 2.1. CONVEX SET 41(a)R(b)R 2(c)R 3F
- Page 43 and 44: 2.1. CONVEX SET 43where Q∈ R 3×3
- Page 45 and 46: 2.1. CONVEX SET 45Now let’s move
- Page 47 and 48: 2.1. CONVEX SET 47By additive inver
- Page 49 and 50: 2.1. CONVEX SET 49R nR mR(A T )x px
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- Page 61 and 62: 2.3. HULLS 61Figure 20: Convex hull
- Page 63 and 64: 2.3. HULLS 63Aaffine hull (drawn tr
- Page 65 and 66: 2.3. HULLS 65subset of the affine h
- Page 67 and 68: 2.3. HULLS 672.3.2.0.2 Example. Nuc
- Page 69 and 70: 2.3. HULLS 692.3.2.0.3 Exercise. Co
- Page 71 and 72: 2.3. HULLS 71Figure 24: A simplicia
- Page 73 and 74: 2.4. HALFSPACE, HYPERPLANE 73H + =
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- Page 85 and 86: 2.5. SUBSPACE REPRESENTATIONS 852.5
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- Page 97 and 98: 2.7. CONES 97X(a)00(b)XFigure 33: (
- Page 99 and 100: 2.7. CONES 99XXFigure 37: Truncated
- Page 101 and 102: 2.7. CONES 101Figure 39: Not a cone
- Page 103 and 104: 2.7. CONES 103cone that is a halfli
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2.4. HALFSPACE, HYPERPLANE 81tradition(a)Yy paH +H −∂H −nontraditional(b)Yy pãH −H +∂H +Figure 29: (a) Hyperplane ∂H − (128) 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 28. (b) Hyperplane ∂H + nontraditionally supporting Y .Vector ã is inward-normal to hyperplane now with respect to bothhalfspace H + and set Y . Tradition [199] [307] recognizes only positivenormal polarity in support function σ Y as in (129); id est, normal a ,figure (a). But both interpretations of supporting hyperplane are useful.
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