v2009.01.01 - Convex Optimization

318 CHAPTER 4. SEMIDEFINITE PROGRAMMING
The trace constraint on X normalizes vector x while the diagonal constraint
on Z maintains sign between respective entries of x and y . Regularization
term ‖X −Y ‖ F then makes x equal to y to within a real scalar; (C.2.0.0.2)
in this case, a positive scalar. To make this program solvable by convex
iteration, as explained in Example 4.4.1.2.2 and other previous examples, we
move the rank constraint to the objective
minimize
X , Y , Z , x , y
subject to (x, y) ∈ C
⎡
f(x, y) + ‖X − Y ‖ F + 〈G, W 〉
G = ⎣
tr(X) = 1
δ(Z) ≽ 0
X Z x
Z Y y
x T y T 1
⎤
⎦≽ 0
(727)
by introducing a direction matrix W found from (1581a):
minimize
W ∈ S 2N+1 〈G ⋆ , W 〉
subject to 0 ≼ W ≼ I
trW = 2N
(728)
This semidefinite program has an optimal solution that is known in
closed form. Iteration (727) (728) terminates when rankG = 1 and linear
regularization 〈G, W 〉 vanishes to within some numerical tolerance in (727);
typically, in two iterations. If function f competes too much with the
regularization, positively weighting each regularization term will become
required. At convergence, problem (727) becomes a convex equivalent to
the original nonconvex problem (724).
4.6.0.0.8 Example. fast max cut. [96]
Let Γ be an n-node graph, and let the arcs (i , j) of the graph be
associated with [ ] weights a ij . The problem is to find a cut of the
largest possible weight, i.e., to partition the set of nodes into two
parts S, S ′ in such a way that the total weight of all arcs linking
S and S ′ (i.e., with one incident node in S and the other one
in S ′ [Figure 85]) is as large as possible. [31,4.3.3]