SOS-SDP: An Exact Solver for Minimum Sum-of-Squares Clustering

@article{Piccialli2021SOSSDPAE,
  title={SOS-SDP: An Exact Solver for Minimum Sum-of-Squares Clustering},
  author={Veronica Piccialli and Antonio M. Sudoso and Angelika Wiegele},
  journal={INFORMS Journal on Computing},
  year={2021}
}
The minimum sum-of-squares clustering problem (MSSC) consists of partitioning n observations into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure in which valid inequalities are iteratively added to the Peng–Wei semidefinite programming (SDP) relaxation. The upper… 
[PDF] Semantic Reader
6 Citations
Background Citations
2
Methods Citations
1

Figures and Tables from this paper

6 Citations

Sketch-and-solve approaches to k-means clustering by semidefinite programming

A sketch-and-solve approach to speed up the Peng-Wei semidefinite relaxation of k -means clustering and certify approximate optimality of clustering solutions obtained by k-means++.

Fix and Bound: An efficient approach for solving large-scale quadratic programming problems with box constraints

A branch-and-bound algorithm for solving nonconvex quadratic programming problems with box constraints (BoxQP) that combines existing tools, such as semidefinite programming (SDP) bounds strengthened through valid inequalities with a new class of optimality-based linear cuts which leads to variable-xing.

Global Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming

This paper proposes an exact approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC, and derives a new SDP relaxation that scales better with the instance size and the number of clusters.

Polynomial Optimization: Enhancing RLT relaxations with Conic Constraints

This research investigates the strengthening of RLT relaxations of polynomial optimization problems through the addition of nine different types of constraints that are based on linear, second-order cone, and semidefinite programming to solve to optimality the instances of well established test sets of polymouth optimization problems.

An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering

Mixed-Integer Nonlinear Programming for State-Based Non-Intrusive Load Monitoring

The proposed formulation is computationally efficient, able to disambiguate loads with similar consumption patterns, and successfully reconstruct the signatures of known appliances despite the presence of unmetered devices, thus overcoming the main drawbacks of the optimization-based methods available in the literature.

References

SHOWING 1-10 OF 66 REFERENCES

A new theoretical framework for K-means-type clustering

The0-1 SDP model can be applied not only to MSSC, but also to other scenarios of clustering as well, and it is shown that the recently proposed normalized k-cut and spectral clustering can also be embedded into the 0-1SDP model in various kernel spaces.

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

This work considers a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods and shows that the positive definiteness of the generalized Hessian of the objective function in these inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems.

Approximating K-means-type Clustering via Semidefinite Programming

This paper first model MSSC as a so-called 0-1 semidefinite programming (SDP) problem, and shows that this model provides a unified framework for several clustering approaches such as normalized k-cut and spectral clustering.

An improved column generation algorithm for minimum sum-of-squares clustering

This work proposes a new way to solve the auxiliary problem of finding a column with negative reduced cost based on geometric arguments that greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2,300 entities.

A branch-and-cut SDP-based algorithm for minimum sum-of-squares clustering

A branch-and-cut algorithm is proposed for the underlying 0-1 SDP model that obtains exact solutions for fairly large data sets with computing times comparable with those of the best exact method found in the literature.

Evaluating a branch-and-bound RLT-based algorithm for minimum sum-of-squares clustering

A reformulation-linearization based branch-and-bound algorithm for minimum sum-of-squares clustering, claiming to solve instances with up to 1,000 points, is investigated in further detail, reproducing some of their computational experiments.

Memetic differential evolution methods for clustering problems

Special issue: Statistical methods & models for the evaluation systems of the public sector

A simulated annealing algorithm with a dual perturbation method for clustering

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

This paper studies the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP and proposes a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm.
...