ADMM for SDP Relaxation of GP
Loading...
Date
2016-08-30
Authors
Sun, Hao
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
We consider the problem of partitioning the set of nodes of a graph G into k sets of
given sizes in order to minimize the cut obtained after removing the k-th set. This is a
variant of the well-known vertex separator problem that has applications in e.g., numerical
linear algebra. This problem is well studied and there are many lower bounds such as:
the standard eigenvalue bound; projected eigenvalue bounds using both the adjacency
matrix and the Laplacian; quadratic programming (QP) bounds derived from imitating
the (QP) bounds for the quadratic assignment problem; and semidefinite programming
(SDP) bounds. For the quadratic assignment problem, a recent paper of [8] had great
success from applying the ADMM (altenating direction method of multipliers) to the SDP
relaxation. We consider the SDP relaxation of the vertex separator problem and the
application of the ADMM method in solving the SDP. The main advantage of the ADMM
method is that optimizing over the set of doubly non-negative matrices is about as difficult
as optimizing over the set of positive semidefinite matrices. Enforcing the non-negativity
constraint gives us a clear improvement in the quality of bounds obtained. We implement
both a high rank and a nonconvex low rank ADMM method, where the difference is the
choice of rank of the projection onto the semidefinite cone. As for the quadratic assignment
problem, though there is no theoretical convergence guarantee, the nonconvex approach
always converges to a feasible solution in practice.
Description
Keywords
Graph Partitioning, alternating direction method of multipliers, Semidefinite Programmming