v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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112 CHAPTER 2. CONVEX GEOMETRYBCADFigure 42: Properties of extreme points carry over to extreme directions.[307,18] Four rays (drawn truncated) on boundary of conic hull oftwo-dimensional closed convex set from Figure 32 lifted to R 3 . Raythrough point A is exposed hence extreme. Extreme direction B on coneboundary is not an exposed direction, although it belongs to the exposedface cone{A,B}. Extreme ray through C is exposed. Point D is neitheran exposed or extreme direction although it belongs to a two-dimensionalexposed face of the conic hull.0

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 113Thus each and every extreme point of a convex set (that is not a point)resides on its relative boundary, while each and every extreme direction of aconvex set (that is not a halfline and contains no line) resides on its relativeboundary because extreme points and directions of such respective sets donot belong to relative interior by definition.The relationship between extreme sets and the relative boundary actuallygoes deeper: Any face F of convex set C (that is not C itself) belongs torel ∂ C , so dim F < dim C . [307,18.1.3]2.8.2.2 Converse caveatIt is inconsequent to presume that each and every extreme point and directionis necessarily exposed, as might be erroneously inferred from the conventionalboundary definition (2.6.1.4.1); 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 24, 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 32,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 42.2.9 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 [26, p.78]

112 CHAPTER 2. CONVEX GEOMETRYBCADFigure 42: Properties of extreme points carry over to extreme directions.[307,18] Four rays (drawn truncated) on boundary of conic hull oftwo-dimensional closed convex set from Figure 32 lifted to R 3 . Raythrough point A is exposed hence extreme. Extreme direction B on coneboundary is not an exposed direction, although it belongs to the exposedface cone{A,B}. Extreme ray through C is exposed. Point D is neitheran exposed or extreme direction although it belongs to a two-dimensionalexposed face of the conic hull.0

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