v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
62 CHAPTER 2. CONVEX GEOMETRYfor which we call list X a set of generators. Hull A is parallel to subspacewhereR{x l − x 1 , l=2... N} = R(X − x 1 1 T ) ⊆ R n (79)R(A) = {Ax | ∀x} (142)Given some arbitrary set C and any x∈ Cwhere aff(C−x) is a subspace.aff C = x + aff(C − x) (80)2.3.1.0.1 Definition. Affine subset.We analogize affine subset to subspace, 2.14 defining it to be any nonemptyaffine set (2.1.4).△The affine hull of a point x is that point itself;aff ∅ ∅ (81)aff{x} = {x} (82)Affine hull of two distinct points is the unique line through them. (Figure 21)The affine hull of three noncollinear points in any dimension is that uniqueplane containing the points, and so on. The subspace of symmetric matricesS m is the affine hull of the cone of positive semidefinite matrices; (2.9)aff S m + = S m (83)2.3.1.0.2 Example. Affine hull of rank-1 correlation matrices. [218]The set of all m ×m rank-1 correlation matrices is defined by all the binaryvectors y in R m (confer5.9.1.0.1){yy T ∈ S m + | δ(yy T )=1} (84)Affine hull of the rank-1 correlation matrices is equal to the set of normalizedsymmetric matrices; id est,aff{yy T ∈ S m + | δ(yy T )=1} = {A∈ S m | δ(A)=1} (85)2.14 The popular term affine subspace is an oxymoron.
2.3. HULLS 63Aaffine hull (drawn truncated)Cconvex hullKconic hull (truncated)Rrange or span is a plane (truncated)Figure 21: Given two points in Euclidean vector space of any dimension,their various hulls are illustrated. Each hull is a subset of range; generally,A , C, K ⊆ R ∋ 0. (Cartesian axes drawn for reference.)
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- Page 21 and 22: Chapter 1OverviewConvex Optimizatio
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- 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
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- Page 61: 2.3. HULLS 61Figure 20: Convex hull
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2.3. HULLS 63Aaffine hull (drawn truncated)Cconvex hullKconic hull (truncated)Rrange or span is a plane (truncated)Figure 21: Given two points in Euclidean vector space of any dimension,their various hulls are illustrated. Each hull is a subset of range; generally,A , C, K ⊆ R ∋ 0. (Cartesian axes drawn for reference.)
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