# 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 J. Comput.}, year={2021}, volume={34}, pages={2144-2162} }

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…

## 8 Citations

### Mixed-integer programming techniques for the minimum sum-of-squares clustering problem

- Computer ScienceJournal of Global Optimization
- 2023

This paper develops and test different tailored mixed-integer programming techniques to improve the performance of state-of-the-art MINLP solvers when applied to the problem, among them are cutting planes, propagation techniques, branching rules, or primal heuristics.

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

- Computer ScienceArXiv
- 2022

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

- Computer Science
- 2022

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

- Computer ScienceArXiv
- 2022

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

- Computer ScienceArXiv
- 2022

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

- Computer ScienceComput. Oper. Res.
- 2022

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

- Engineering, Computer ScienceIEEE Transactions on Smart Grid
- 2022

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

- Computer Science
- 2004

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

- MathematicsSIAM J. Optim.
- 2010

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

- Computer ScienceSIAM J. Optim.
- 2007

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

- Computer ScienceMath. Program.
- 2012

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

- Computer Science
- 2008

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

- Computer ScienceJ. Glob. Optim.
- 2011

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

- Computer SciencePattern Recognit.
- 2021

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

- Business
- 2021

### A simulated annealing algorithm with a dual perturbation method for clustering

- Computer SciencePattern Recognit.
- 2021

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

- Computer ScienceINFORMS Journal on Computing
- 2021

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.