v2010.10.26 - Convex Optimization

180 CHAPTER 2. CONVEX GEOMETRY2.13.4.3.3 Exercise. Smallest face of positive semidefinite cone.Derive (221) from (372).2.13.5 Dual PSD cone and generalized inequalityThe dual positive semidefinite cone K ∗ is confined to S M by convention;S M ∗+ {Y ∈ S M | 〈Y , X〉 ≥ 0 for all X ∈ S M + } = S M + (377)The positive semidefinite cone is selfdual in the ambient space of symmetricmatrices [61, exmp.2.24] [39] [195,II]; K = K ∗ .Dual 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 (378)Membership to this cone can be determined in the isometrically isomorphicEuclidean space R M2 via (38). (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 (378):X ≽ 0 ⇔ 〈yy T , X〉 ≥ 0 ∀yy T (≽ 0) (1442)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 (2.9.2.7).2.13.5.1 selfdual conesFrom (131) (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 [199,C.2.3.1],or from discretized definition (369) of the dual cone we get a ratherself-evident characterization of selfdual cones: