Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped. This paper leverages a minimum rank completion result to decompose the rank constraint on a single large matrix into multiple rank constraints on a set of smaller matrices. The re-weighted heuristic is used as a proxy for rank, and the specific form of the heuristic preserves the sparsity pattern between iterations. Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed. The problem of subspace clustering is used to demonstrate the computational improvement of the proposed method.Comment: 6 pages, 6 figure

Solution of Optimal Power Flow Problems using Moment Relaxations Augmented with Objective Function Penalization

The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of "moment" relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate low-impedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems.Comment: 8 pages, 1 figure, to appear in IEEE 54th Annual Conference on Decision and Control (CDC), 15-18 December 201