v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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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]

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 319 2 3 4 S 1 CUT 16 5 S ′ 15 4 -1 -3 5 1 -3 6 7 2 1 14 13 9 8 12 11 10 Figure 85: A cut partitions nodes {i=1...16} of this graph into S and S ′ . Linear arcs have circled weights. The problem is to find a cut maximizing total weight of all arcs linking partitions made by the cut.

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 319<br />

2 3 4<br />

S<br />

1<br />

CUT<br />

16<br />

5<br />

S ′<br />

15<br />

4<br />

-1<br />

-3<br />

5<br />

1<br />

-3<br />

6<br />

7<br />

2<br />

1<br />

14<br />

13<br />

9<br />

8<br />

12<br />

11<br />

10<br />

Figure 85: A cut partitions nodes {i=1...16} of this graph into S and S ′ .<br />

Linear arcs have circled weights. The problem is to find a cut maximizing<br />

total weight of all arcs linking partitions made by the cut.

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