v2007.09.13 - Convex Optimization

56 CHAPTER 2. CONVEX GEOMETRY2.3.1.0.2 Example. Affine hull of rank-1 correlation matrices. [159]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} (73)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} (74)2.3.1.0.3 Exercise. Affine hull of correlation matrices.Prove (74) via definition of affine hull. Find the convex hull instead. 2.3.1.1 Comparison with respect to R N + and S M +The notation a ≽ 0 means vector a belongs to the nonnegative orthantR N + , whereas a ≽ b denotes comparison of vector a to vector b on R Nwith respect to the nonnegative orthant; id est, a ≽ b means a −b belongsto the nonnegative orthant, but neither a or b necessarily belongs to thatorthant. In particular, a ≽ b ⇔ a i ≽ b i ∀i. (320)The symbol ≥ is reserved for scalar comparison on the real line R withrespect to the nonnegative real line R + as in a T y ≥ b . Comparison ofmatrices with respect to the positive semidefinite cone S M + , like I ≽A ≽ 0in Example 2.3.2.0.1, is explained in2.9.0.1.2.3.2 Convex hullThe convex hull [147,A.1.4] [46,2.1.4] [228] of any bounded 2.15 list (or set)of N points X ∈ R n×N forms a unique convex polyhedron (2.12.0.0.1) whosevertices constitute some subset of that list;P ∆ = conv {x l , l=1... N} = conv X = {Xa | a T 1 = 1, a ≽ 0} ⊆ R n(75)2.15 A set in R n is bounded if and only if it can be contained in a Euclidean ball havingfinite radius. [77,2.2] (confer5.7.3.0.1)