v2007.09.13 - Convex Optimization

154 CHAPTER 2. CONVEX GEOMETRYDual generalized inequalities with respect to the positive semidefinite conein the ambient space of symmetric matrices can therefore be simply stated:(Fejér)X ≽ 0 ⇔ tr(Y T X) ≥ 0 for all Y ≽ 0 (322)Membership to this cone can be determined in the isometrically isomorphicEuclidean space R M2 via (31). (2.2.1) By the two interpretations in2.13.1,positive semidefinite matrix Y can be interpreted as inward-normal to ahyperplane supporting the positive semidefinite cone.The fundamental statement of positive semidefiniteness, y T Xy ≥0 ∀y(A.3.0.0.1), evokes a particular instance of these dual generalizedinequalities (322):X ≽ 0 ⇔ 〈yy T , X〉 ≥ 0 ∀yy T (≽ 0) (1241)Discretization (2.13.4.2.1) allows replacement of positive semidefinitematrices Y with this minimal set of generators comprising the extremedirections of the positive semidefinite cone (2.9.2.4).2.13.5.1 self-dual conesFrom (110) (a consequence of the halfspaces theorem (2.4.1.1.1)), where theonly finite value of the support function for a convex cone is 0 [147,C.2.3.1],or from discretized definition (319) of the dual cone we get a ratherself-evident characterization of self-duality:K = K ∗ ⇔ K = ⋂ {y | γ T y ≥ 0 } (323)γ∈G(K)In words: Cone K is self-dual iff its own extreme directions areinward-normals to a (minimal) set of hyperplanes bounding halfspaces whoseintersection constructs it. This means each extreme direction of K is normalto a hyperplane exposing one of its own faces; a necessary but insufficientcondition for self-duality (Figure 48, for example).Self-dual cones are of necessarily nonempty interior [25,I] and invariantto rotation about the origin. Their most prominent representatives are theorthants, the positive semidefinite cone S M + in the ambient space of symmetricmatrices (321), and the Lorentz cone (147) [17,II.A] [46, exmp.2.25]. Inthree dimensions, a plane containing the axis of revolution of a self-dual cone(and the origin) will produce a slice whose boundary makes a right angle.