v2007.11.26 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 289move the rank constraint to the objectiveminimizeX , Y , Z , x , ysubject to (x, y) ∈ C⎡f(x, y) + ‖X − Y ‖ F + 〈G, Y 〉G = ⎣tr(X) = 1δ(Z) ≽ 0X Z xZ Y yx T y T 1⎤⎦≽ 0(688)by introducing a direction matrix Y found from (1506a)minimizeY ∈ S 2N+1 〈G ⋆ , Y 〉subject to 0 ≼ Y ≼ ItrY = 2N(689)which has an optimal solution that is known in closed form. Iteration(688) (689) terminates when rankG = 1 and regularization 〈G, Y 〉 vanishesto within some numerical tolerance in (688); typically, in two iterations.If function f competes too much with the regularization, positivelyweighting each regularization term will become required. At convergence,problem (688) becomes a convex equivalent to the original nonconvexproblem (685).4.6.0.0.6 Example. fast max cut. [77]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 76]) is as large as possible. [27,4.3.3]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 [83], quantumcomputing [297].