v2009.01.01 - Convex Optimization

160 CHAPTER 2. CONVEX GEOMETRY
2.13.4.2.2 Exercise. Test of discretized dual generalized inequalities.
Test Theorem 2.13.4.2.1 on Figure 49(a) using the extreme directions as
generators.
From the discretized membership theorem we may further deduce a more
surgical description of closed convex cone that prescribes only a finite number
of halfspaces for its construction when polyhedral: (Figure 48(a))
K = {x ∈ R n | 〈γ ∗ , x〉 ≥ 0 for all γ ∗ ∈ G(K ∗ )} (331)
K ∗ = {y ∈ R n | 〈γ , y〉 ≥ 0 for all γ ∈ G(K)} (332)
2.13.4.2.3 Exercise. Partial order induced by orthant.
When comparison is with respect to the nonnegative orthant K = R n + , then
from the discretized membership theorem it directly follows:
x ≼ z ⇔ x i ≤ z i ∀i (333)
Generate simple counterexamples demonstrating that this equivalence with
entrywise inequality holds only when the underlying cone inducing partial
order is the nonnegative orthant.
2.13.5 Dual PSD cone and generalized inequality
The 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 + (334)
The positive semidefinite cone is self-dual in the ambient space of symmetric
matrices [53, exmp.2.24] [35] [170,II]; K = K ∗ .
Dual generalized inequalities with respect to the positive semidefinite cone
in the ambient space of symmetric matrices can therefore be simply stated:
(Fejér)
X ≽ 0 ⇔ tr(Y T X) ≥ 0 for all Y ≽ 0 (335)
Membership to this cone can be determined in the isometrically isomorphic
Euclidean 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 a
hyperplane supporting the positive semidefinite cone.