3,922 research outputs found
An interior point algorithm for minimum sum-of-squares clustering
Copyright @ 2000 SIAM PublicationsAn exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the rst time for several fairly large data sets from the literature, including Fisher's 150 iris.This research was supported by the Fonds
National de la Recherche Scientifique Suisse, NSERC-Canada, and FCAR-Quebec
Construction of power flow feasibility sets
We develop a new approach for construction of convex analytically simple
regions where the AC power flow equations are guaranteed to have a feasible
solutions. Construction of these regions is based on efficient semidefinite
programming techniques accelerated via sparsity exploiting algorithms.
Resulting regions have a simple geometric shape in the space of power
injections (polytope or ellipsoid) and can be efficiently used for assessment
of system security in the presence of uncertainty. Efficiency and tightness of
the approach is validated on a number of test networks
Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time
We give a nearly linear time randomized approximation scheme for the
Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an
undirected edge-weighted graph on edges and , the
algorithm outputs in time, with high probability, a
-approximation to the Held-Karp bound on the metric TSP instance
induced by the shortest path metric on . The algorithm can also be used to
output a corresponding solution to the Subtour Elimination LP. We substantially
improve upon the running time achieved previously
by Garg and Khandekar. The LP solution can be used to obtain a fast randomized
-approximation for metric TSP which improves
upon the running time of previous implementations of Christofides' algorithm
Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming
Elfving's Theorem is a major result in the theory of optimal experimental
design, which gives a geometrical characterization of optimality. In this
paper, we extend this theorem to the case of multiresponse experiments, and we
show that when the number of experiments is finite, and optimal
design of multiresponse experiments can be computed by Second-Order Cone
Programming (SOCP). Moreover, our SOCP approach can deal with design problems
in which the variable is subject to several linear constraints.
We give two proofs of this generalization of Elfving's theorem. One is based
on Lagrangian dualization techniques and relies on the fact that the
semidefinite programming (SDP) formulation of the multiresponse optimal
design always has a solution which is a matrix of rank . Therefore, the
complexity of this problem fades.
We also investigate a \emph{model robust} generalization of optimality,
for which an Elfving-type theorem was established by Dette (1993). We show with
the same Lagrangian approach that these model robust designs can be computed
efficiently by minimizing a geometric mean under some norm constraints.
Moreover, we show that the optimality conditions of this geometric programming
problem yield an extension of Dette's theorem to the case of multiresponse
experiments.
When the number of unknown parameters is small, or when the number of linear
functions of the parameters to be estimated is small, we show by numerical
examples that our approach can be between 10 and 1000 times faster than the
classic, state-of-the-art algorithms
- ā¦