v2009.01.01 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 121
2.9.2.5.3 Exercise. Dual inscription.
Find dual proper polyhedral cone K ∗ from Figure 42.
2.9.2.6 Boundary constituents of the positive semidefinite cone
2.9.2.6.1 Lemma. Sum of positive semidefinite matrices.
For A,B ∈ S M +
rank(A + B) = rank(µA + (1 −µ)B) (229)
over the open interval (0, 1) of µ .
Proof. Any positive semidefinite matrix belonging to the PSD cone
has an eigen decomposition that is a positively scaled sum of linearly
independent symmetric dyads. By the linearly independent dyads definition
inB.1.1.0.1, rank of the sum A +B is equivalent to the number of linearly
independent dyads constituting it. Linear independence is insensitive to
further positive scaling by µ . The assumption of positive semidefiniteness
prevents annihilation of any dyad from the sum A +B .
2.9.2.6.2 Example. Rank function quasiconcavity. (confer3.3)
For A,B ∈ R m×n [176,0.4]
that follows from the fact [287,3.6]
rankA + rankB ≥ rank(A + B) (230)
dim R(A) + dim R(B) = dim R(A + B) + dim(R(A) ∩ R(B)) (231)
For A,B ∈ S M +
rankA + rankB ≥ rank(A + B) ≥ min{rankA, rankB} (1370)
that follows from the fact
N(A + B) = N(A) ∩ N(B) , A,B ∈ S M + (146)
Rank is a quasiconcave function on S M + because the right-hand inequality in
(1370) has the concave form (578); videlicet, Lemma 2.9.2.6.1.
From this example we see, unlike convex functions, quasiconvex functions
are not necessarily continuous. (3.3) We also glean:
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