v2009.01.01 - Convex Optimization

522 CHAPTER 7. PROXIMITY PROBLEMS
To see that, substitute the nonincreasingly ordered diagonalizations
Y i + εI ∆ = Q(Λ + εI)Q T
Y ∆ = UΥU T
(a)
(b)
(1278)
into (1277). Then from (1582) we have,
inf δ((Λ + εI) −1 ) T δ(Υ) =
Υ∈U ⋆T CU ⋆
≤
inf
Υ∈U T CU
inf tr ( (Λ + εI) −1 R T ΥR )
R T =R −1
inf tr((Y i + εI) −1 Y )
Y ∈ C
(1279)
where R = ∆ Q T U in U on the set of orthogonal matrices is a bijection. The
role of ε is, therefore, to limit the maximum weight; the smallest entry on
the main diagonal of Υ gets the largest weight.
7.2.2.5 Applying log det rank-heuristic to Problem 2
When the log det rank-heuristic is inserted into Problem 2, problem (1273)
becomes the problem sequence in i
minimize
D , Y , κ
subject to
κρ + 2 tr(V Y V )
[ ]
djl y jl
≽ 0 ,
y jl
h 2 jl
l > j = 1... N −1
− tr((−V D i V + εI) −1 V DV ) ≤ κρ
(1280)
Y ∈ S N h
D ∈ EDM N
where D i+1 ∆ =D ⋆ ∈ EDM N , and D 0 ∆ =11 T − I .
7.2.2.6 Tightening this log det rank-heuristic
Like the trace method, this log det technique for constraining rank offers
no provision for meeting a predetermined upper bound ρ . Yet since the
eigenvalues of the sum are simply determined, λ(Y i + εI) = δ(Λ + εI) , we
may certainly force selected weights to ε −1 by manipulating diagonalization
(1278a). Empirically we find this sometimes leads to better results, although
affine dimension of a solution cannot be guaranteed.