v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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114 CHAPTER 2. CONVEX GEOMETRY For two-dimensional matrices (M =2, Figure 37) {yy T ∈ S 2 | y ∈ R 2 } = ∂S 2 + (208) while for one-dimensional matrices, in exception, (M =1,2.7) {yy ∈ S | y≠0} = int S + (209) Each and every extreme direction yy T makes the same angle with the identity matrix in isomorphic R M(M+1)/2 , dependent only on dimension; videlicet, 2.37 ( ) (yy T 〈yy T , I〉 1 , I) = arccos = arccos √ ∀y ∈ R M (210) ‖yy T ‖ F ‖I‖ F M 2.9.2.4.1 Example. Positive semidefinite matrix from extreme directions. Diagonalizability (A.5) of symmetric matrices yields the following results: Any symmetric positive semidefinite matrix (1353) can be written in the form A = ∑ λ i z i zi T = ÂÂT = ∑ â i â T i ≽ 0, λ ≽ 0 (211) i i a conic combination of linearly independent extreme directions (â i â T i or z i z T i where ‖z i ‖=1), where λ is a vector of eigenvalues. If we limit consideration to all symmetric positive semidefinite matrices bounded such that trA=1 C ∆ = {A ≽ 0 | trA = 1} (212) then any matrix A from that set may be expressed as a convex combination of linearly independent extreme directions; A = ∑ i λ i z i z T i ∈ C , 1 T λ = 1, λ ≽ 0 (213) Implications are: 1. set C is convex, (it is an intersection of PSD cone with hyperplane) 2. because the set of eigenvalues corresponding to a given square matrix A is unique, no single eigenvalue can exceed 1 ; id est, I ≽ A . Set C is an instance of Fantope (83). 2.37 Analogy with respect to the EDM cone is considered in [159, p.162] where it is found: angle is not constant. Extreme directions of the EDM cone can be found in6.5.3.2. The cone’s axis is −E =11 T − I (1000).

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 115 R 0 Figure 40: This circular cone continues upward infinitely. Axis of revolution is illustrated as vertical line segment through origin. R is the radius, the distance measured from any extreme direction to axis of revolution. Were this a Lorentz cone, any plane slice containing the axis of revolution would make a right angle.

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 115<br />

R<br />

0<br />

Figure 40: This circular cone continues upward infinitely. Axis of revolution<br />

is illustrated as vertical line segment through origin. R is the radius, the<br />

distance measured from any extreme direction to axis of revolution. Were<br />

this a Lorentz cone, any plane slice containing the axis of revolution would<br />

make a right angle.

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