v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
40 CHAPTER 2. CONVEX GEOMETRY(a)(b)R 2(c)Figure 12: (a) Closed convex set. (b) Neither open, closed, or convex. YetPSD cone can remain convex in absence of certain boundary components(2.9.2.9.3). Nonnegative orthant with origin excluded (2.6) and positiveorthant with origin adjoined [307, p.49] are convex. (c) Open convex set.Given the intersection of convex set C with affine set Arel int(C ∩ A) = rel int(C) ∩ A ⇐ rel int(C) ∩ A ̸= ∅ (15)Because an affine set A is openrel int A = A (16)2.1.7 classical boundary(confer2.1.7.2) Boundary of a set C is the closure of C less its interior;∂ C = C \ int C (17)[55,1.1] which follows from the factint C = C ⇔ ∂ int C = ∂ C (18)and presumption of nonempty interior. 2.8 Implications are:2.8 Otherwise, for x∈ R n as in (12), [258,2.1-2.3]int{x} = ∅ = ∅
2.1. CONVEX SET 41(a)R(b)R 2(c)R 3Figure 13: (a) Ellipsoid in R is a line segment whose boundary comprises twopoints. Intersection of line with ellipsoid in R , (b) in R 2 , (c) in R 3 . Eachellipsoid illustrated has entire boundary constituted by zero-dimensionalfaces; in fact, by vertices (2.6.1.0.1). Intersection of line with boundaryis a point at entry to interior. These same facts hold in higher dimension.int C = C \∂ Ca bounded open set has boundary defined but not contained in the setinterior of an open set is equivalent to the set itself;from which an open set is defined: [258, p.109]C is open ⇔ int C = C (19)C is closed ⇔ int C = C (20)The set illustrated in Figure 12b is not open because it is not equivalentto its interior, for example, it is not closed because it does not containits boundary, and it is not convex because it does not contain all convexcombinations of its boundary points.the empty set is both open and closed.
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40 CHAPTER 2. CONVEX GEOMETRY(a)(b)R 2(c)Figure 12: (a) Closed convex set. (b) Neither open, closed, or convex. YetPSD cone can remain convex in absence of certain boundary components(2.9.2.9.3). Nonnegative orthant with origin excluded (2.6) and positiveorthant with origin adjoined [307, p.49] are convex. (c) Open convex set.Given the intersection of convex set C with affine set Arel int(C ∩ A) = rel int(C) ∩ A ⇐ rel int(C) ∩ A ̸= ∅ (15)Because an affine set A is openrel int A = A (16)2.1.7 classical boundary(confer2.1.7.2) Boundary of a set C is the closure of C less its interior;∂ C = C \ int C (17)[55,1.1] which follows from the factint C = C ⇔ ∂ int C = ∂ C (18)and presumption of nonempty interior. 2.8 Implications are:2.8 Otherwise, for x∈ R n as in (12), [258,2.1-2.3]int{x} = ∅ = ∅
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