v2009.01.01 - Convex Optimization

518 CHAPTER 7. PROXIMITY PROBLEMS
Confinement of G to the geometric center subspace provides numerical
stability and no loss of generality (confer (1092)); implicit constraint G1 = 0
is otherwise unnecessary.
To include constraints on the list X ∈ R n×N , we would first rewrite (1265)
minimize − tr(V (D(G) − 2Y )V )
G∈S N c , Y ∈ S N h , X∈ Rn×N [ ]
〈Φij , G〉 y ij
subject to
≽ 0 ,
y ij
h 2 ij
[ ] I X
X T ≽ 0
G
X ∈ C
N ≥ j > i = 1... N −1
(1266)
and then add the constraints, realized here in abstract membership to some
convex set C . This problem realization includes a convex relaxation of the
nonconvex constraint G = X T X and, if desired, more constraints on G could
be added. This technique is discussed in5.4.2.2.4.
7.2.2 Minimization of affine dimension in Problem 2
When desired affine dimension ρ is diminished, the rank function becomes
reinserted into problem (1260) that is then rendered difficult to solve because
the feasible set {D , Y } loses convexity in S N h × R N×N . Indeed, the rank
function is quasiconcave (3.3) on the positive semidefinite cone; (2.9.2.6.2)
id est, its sublevel sets are not convex.
7.2.2.1 Rank minimization heuristic
A remedy developed in [113] [226] [114] [112] introduces convex envelope
(cenv) of the quasiconcave rank function: (Figure 130)
7.2.2.1.1 Definition. Convex envelope. [172]
The convex envelope of a function f : C →R is defined as the largest convex
function g such that g ≤ f on convex domain C ⊆ R n . 7.15 △
7.15 Provided f ≢+∞ and there exists an affine function h ≤f on R n , then the convex
envelope is equal to the convex conjugate (the Legendre-Fenchel transform) of the convex
conjugate of f ; id est, the conjugate-conjugate function f ∗∗ . [173,E.1]