v2010.10.26 - Convex Optimization

64 CHAPTER 2. CONVEX GEOMETRY2.3.1.0.3 Exercise. Affine hull of correlation matrices.Prove (85) via definition of affine hull. Find the convex hull instead. 2.3.1.1 Partial order induced by R N + and S M +Notation a ≽ 0 means vector a belongs to nonnegative orthant R N + whilea ≻ 0 means vector a belongs to the nonnegative orthant’s interior int R N + .a ≽ b denotes comparison of vector a to vector b on R N with respect to thenonnegative orthant; id est, a ≽ b means a −b belongs to the nonnegativeorthant but neither a or b is necessarily nonnegative. With particularrespect to the nonnegative orthant, a ≽ b ⇔ a i ≥ b i ∀i (370).More generally, a ≽ Kb or a ≻ Kb denotes comparison with respect topointed closed convex cone K , but equivalence with entrywise comparisondoes not hold and neither a or b necessarily belongs to K . (2.7.2.2)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 [199,A.1.4] [307] of any bounded 2.15 list or set of N pointsX ∈ R n×N forms a unique bounded convex polyhedron (confer2.12.0.0.1)whose vertices constitute some subset of that list;P conv {x l , l=1... N}= conv X = {Xa | a T 1 = 1, a ≽ 0} ⊆ R n(86)Union of relative interior and relative boundary (2.1.7.2) of the polyhedroncomprise its convex hull P , the smallest closed convex set that contains thelist X ; e.g., Figure 20. Given P , the generating list {x l } is not unique.But because every bounded polyhedron is the convex hull of its vertices,[330,2.12.2] the vertices of P comprise a minimal set of generators.Given some arbitrary set C ⊆ R n , its convex hull conv C is equivalent tothe smallest convex set containing it. (confer2.4.1.1.1) The convex hull is a2.15 An arbitrary set C in R n is bounded iff it can be contained in a Euclidean ball havingfinite radius. [113,2.2] (confer5.7.3.0.1) The smallest ball containing C has radiusinf sup ‖x −y‖ and center x ⋆ whose determination is a convex problem because sup‖x −y‖x y∈Cy∈Cis a convex function of x; but the supremum may be difficult to ascertain.