v2009.01.01 - Convex Optimization

516 CHAPTER 7. PROXIMITY PROBLEMS
7.2.1.1 Equivalent semidefinite program, Problem 2, convex case
Convex problem (1257) is numerically solvable for its global minimum
using an interior-point method [53,11] [342] [251] [239] [334] [10] [121].
We translate (1257) to an equivalent semidefinite program (SDP) for a
pedagogical reason made clear in7.2.2.2 and because there exist readily
available computer programs for numerical solution [141] [312] [31] [335].
Substituting a new matrix variable Y ∆ = [y ij ]∈ R N×N
+
h ij
√
dij ← y ij (1259)
Boyd proposes: problem (1257) is equivalent to the semidefinite program
∑
minimize d ij − 2y ij + h 2 ij
D , Y
i,j
[ ]
dij y ij
(1260)
subject to
≽ 0 , i,j =1... N
y ij h 2 ij
D ∈ EDM N
To see that, recall d ij ≥ 0 is implicit to D ∈ EDM N (5.8.1, (817)). So
when H ∈ R N×N
+ is nonnegative as assumed,
[ ]
dij y √
ij
≽ 0 ⇔ h ij
√d ij ≥ yij 2 (1261)
y ij
h 2 ij
Minimization of the objective function implies maximization of y ij that is
bounded above. √Hence nonnegativity of y ij is implicit to (1260) and, as
desired, y ij →h ij dij as optimization proceeds.
If the given matrix H is now assumed symmetric and nonnegative,
H = [h ij ] ∈ S N ∩ R N×N
+ (1262)
then Y = H ◦ ◦√ D must belong to K= S N h ∩ R N×N
+ (1206). Because Y ∈ S N h
(B.4.2 no.20), then
‖ ◦√ D − H‖ 2 F = ∑ i,j
d ij − 2y ij + h 2 ij = −N tr(V (D − 2Y )V ) + ‖H‖ 2 F (1263)