v2007.09.17 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 281move 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(673)by introducing a direction matrix Y found from (1480a)minimizeY ∈ S 2N+1 〈G ⋆ , Y 〉subject to 0 ≼ Y ≼ ItrY = 2N(674)whose optimal solution has closed form. Iteration (673) (674) terminateswhen rankG = 1 and regularization 〈G, Y 〉 vanishes to within somenumerical tolerance in (673); typically, in two iterations. If function fcompetes too much with the regularization, positively weighting eachregularization term will become required. At convergence, problem (673)becomes a convex equivalent to the original nonconvex problem (670). 4.4.3.0.7 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 ′ ) 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 [295].