8 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
The Boosted DC Algorithm for Clustering with Constraints
This paper aims to investigate the effectiveness of the recently proposed
Boosted Difference of Convex functions Algorithm (BDCA) when applied to
clustering with constraints and set clustering with constraints problems. This
is the first paper to apply BDCA to a problem with nonlinear constraints. We
present the mathematical basis for the BDCA and Difference of Convex functions
Algorithm (DCA), along with a penalty method based on distance functions. We
then develop algorithms for solving these problems and computationally
implement them, with publicly available implementations. We compare old
examples and provide new experiments to test the algorithms. We find that the
BDCA method converges in fewer iterations than the corresponding DCA-based
method. In addition, BDCA yields faster CPU running-times in all tested
problems
Exact algorithms for minimum sum-of-squares clustering
NP-Hardness of Euclidean sum-of-squares clustering -- Computational complexity -- An incorrect reduction from the K-section problem -- A new proof by reduction from the densest cut problem -- Evaluating a branch-and-bound RLT-based algorithm for minimum sum-of-squares clustering -- Reformulation-Linearization technique for the MSSC -- Branch-and-bound for the MSSC -- An attempt at reproducting computational results -- Breaking symmetry and convex hull inequalities -- A branch-and-cut SDP-based algorithm for minimum sum-of-squares clustering -- Equivalence of MSSC to 0-1 SDP -- A branch-and cut algorithm for the 0-1 SDP formulation -- Computational experiments -- An improved column generation algorithm for minimum sum-of-squares clustering -- Column generation algorithm revisited -- A geometric approach -- Generalization to the Euclidean space -- Computational results
A new efficient algorithm based on DC programming and DCA for clustering
International audienc