v2010.10.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 365This semidefinite program has an optimal solution that is known inclosed form. Iteration (816) (817) terminates when rankG = 1 and linearregularization 〈G, W 〉 vanishes to within some numerical tolerance in (816);typically, in two iterations. If function f competes too much with theregularization, positively weighting each regularization term will becomerequired. At convergence, problem (816) becomes a convex equivalent tothe original nonconvex problem (813).4.6.0.0.10 Example. fast max cut. [113]Let Γ be an n-node graph, and let the arcs (i , j) of the graph beassociated with [ ] weights a ij . The problem is to find a cut of thelargest possible weight, i.e., to partition the set of nodes into twoparts S, S ′ in such a way that the total weight of all arcs linkingS and S ′ (i.e., with one incident node in S and the other onein S ′ [Figure 106]) is as large as possible. [34,4.3.3]Literature on the max cut problem is vast because this problem has elegantprimal and dual formulation, its solution is very difficult, and there existmany commercial applications; e.g., semiconductor design [125], quantumcomputing [387].Our purpose here is to demonstrate how iteration of two simple convexproblems can quickly converge to an optimal solution of the max cutproblem with a 98% success rate, on average. 4.39 max cut is stated:∑maximize a ij (1 − x i x j ) 1x21≤i