26 research outputs found
On the Global Linear Convergence of the ADMM with Multi-Block Variables
The alternating direction method of multipliers (ADMM) has been widely used
for solving structured convex optimization problems. In particular, the ADMM
can solve convex programs that minimize the sum of convex functions with
-block variables linked by some linear constraints. While the convergence of
the ADMM for was well established in the literature, it remained an open
problem for a long time whether or not the ADMM for is still
convergent. Recently, it was shown in [3] that without further conditions the
ADMM for may actually fail to converge. In this paper, we show that
under some easily verifiable and reasonable conditions the global linear
convergence of the ADMM when can still be assured, which is important
since the ADMM is a popular method for solving large scale multi-block
optimization models and is known to perform very well in practice even when
. Our study aims to offer an explanation for this phenomenon
Semidefinite Programming Approach for the Quadratic Assignment Problem with a Sparse Graph
The matching problem between two adjacency matrices can be formulated as the
NP-hard quadratic assignment problem (QAP). Previous work on semidefinite
programming (SDP) relaxations to the QAP have produced solutions that are often
tight in practice, but such SDPs typically scale badly, involving matrix
variables of dimension where n is the number of nodes. To achieve a speed
up, we propose a further relaxation of the SDP involving a number of positive
semidefinite matrices of dimension no greater than the number
of edges in one of the graphs. The relaxation can be further strengthened by
considering cliques in the graph, instead of edges. The dual problem of this
novel relaxation has a natural three-block structure that can be solved via a
convergent Augmented Direction Method of Multipliers (ADMM) in a distributed
manner, where the most expensive step per iteration is computing the
eigendecomposition of matrices of dimension . The new SDP
relaxation produces strong bounds on quadratic assignment problems where one of
the graphs is sparse with reduced computational complexity and running times,
and can be used in the context of nuclear magnetic resonance spectroscopy (NMR)
to tackle the assignment problem.Comment: 31 page