20,844 research outputs found
A DC Programming Approach for Solving Multicast Network Design Problems via the Nesterov Smoothing Technique
This paper continues our effort initiated in [9] to study Multicast
Communication Networks, modeled as bilevel hierarchical clustering problems, by
using mathematical optimization techniques. Given a finite number of nodes, we
consider two different models of multicast networks by identifying a certain
number of nodes as cluster centers, and at the same time, locating a particular
node that serves as a total center so as to minimize the total transportation
cost through the network. The fact that the cluster centers and the total
center have to be among the given nodes makes this problem a discrete
optimization problem. Our approach is to reformulate the discrete problem as a
continuous one and to apply Nesterov smoothing approximation technique on the
Minkowski gauges that are used as distance measures. This approach enables us
to propose two implementable DCA-based algorithms for solving the problems.
Numerical results and practical applications are provided to illustrate our
approach
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
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