v2007.09.13 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 97boundary definition (2.6.1.3.1); although it can correctly be inferred: eachand every extreme point and direction belongs to some exposed face.Arbitrary points residing on the relative boundary of a convex set are notnecessarily exposed or extreme points. Similarly, the direction of an arbitraryray, base 0, on the boundary of a convex cone is not necessarily an exposedor extreme direction. For the polyhedral cone illustrated in Figure 15, forexample, there are three two-dimensional exposed faces constituting theentire boundary, each composed of an infinity of rays. Yet there are onlythree exposed directions.Neither is an extreme direction on the boundary of a pointed convex conenecessarily an exposed direction. Lift the two-dimensional set in Figure 21,for example, into three dimensions such that no two points in the set arecollinear with the origin. Then its conic hull can have an extreme directionB on the boundary that is not an exposed direction, illustrated in Figure 30.2.9 Positive semidefinite (PSD) coneThe cone of positive semidefinite matrices studied in this sectionis arguably the most important of all non-polyhedral cones whosefacial structure we completely understand.−Alexander Barvinok [20, p.78]2.9.0.0.1 Definition. Positive semidefinite cone.The set of all symmetric positive semidefinite matrices of particulardimension M is called the positive semidefinite cone:S M ∆+ = { A ∈ S M | A ≽ 0 }= { A ∈ S M | y T Ay ≥0 ∀ ‖y‖ = 1 }= ⋂ {A ∈ S M | 〈yy T , A〉 ≥ 0 } (160)‖y‖=1formed by the intersection of an infinite number of halfspaces (2.4.1.1) invectorized variable A , 2.29 each halfspace having partial boundary containingthe origin in isomorphic R M(M+1)/2 . It is a unique immutable proper conein the ambient space of symmetric matrices S M .2.29 infinite in number when M >1. Because y T A y=y T A T y , matrix A is almost alwaysassumed symmetric. (A.2.1)