v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 161
The fundamental statement of positive semidefiniteness, y T Xy ≥0 ∀y
(A.3.0.0.1), evokes a particular instance of these dual generalized
inequalities (335):
X ≽ 0 ⇔ 〈yy T , X〉 ≥ 0 ∀yy T (≽ 0) (1345)
Discretization (2.13.4.2.1) allows replacement of positive semidefinite
matrices Y with this minimal set of generators comprising the extreme
directions of the positive semidefinite cone (2.9.2.4).
2.13.5.1 self-dual cones
From (122) (a consequence of the halfspaces theorem,2.4.1.1.1), where the
only finite value of the support function for a convex cone is 0 [173,C.2.3.1],
or from discretized definition (332) of the dual cone we get a rather
self-evident characterization of self-duality:
K = K ∗ ⇔ K = ⋂ {
y | γ T y ≥ 0 } (336)
γ∈G(K)
In words: Cone K is self-dual iff its own extreme directions are
inward-normals to a (minimal) set of hyperplanes bounding halfspaces whose
intersection constructs it. This means each extreme direction of K is normal
to a hyperplane exposing one of its own faces; a necessary but insufficient
condition for self-duality (Figure 54, for example).
Self-dual cones are of necessarily nonempty interior [29,I] and invariant
to rotation about the origin. Their most prominent representatives are the
orthants, the positive semidefinite cone S M + in the ambient space of symmetric
matrices (334), and the Lorentz cone (160) [19,II.A] [53, exmp.2.25]. In
three 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.
2.13.5.1.1 Example. Linear matrix inequality. (confer2.13.2.0.3)
Consider a peculiar vertex-description for a closed convex cone defined
over a positive semidefinite cone (instead of a nonnegative orthant as in
definition (94)): for X ∈ S n given A j ∈ S n , j =1... m