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40102 Applied Mathematics
40103 Numerical and Computational Mathematics
30802 Computation Theory and Mathematics
3Nonconvex optimization
20801 Artificial Intelligence and Image Processing
20806 Information Systems
2DC programming
2Incremental algorithm
2Regression analysis
2Similarity measure
2Smoothing techniques
10906 Electrical and Electronic Engineering
1Australian Digital Thesis
1Bundle methods
1K-means algorithm
1Limited memory methods
1Mathematical optimization
1Nonlinear programming

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An incremental nonsmooth optimization algorithm for clustering using L1 and L∞ norms

- Ordin, Burak, Bagirov, Adil, Mohebi, Ehsam

**Authors:**Ordin, Burak , Bagirov, Adil , Mohebi, Ehsam**Date:**2020**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 16, no. 6 (2020), p. 2757-2779**Relation:**DP190100580**Full Text:**false**Reviewed:****Description:**An algorithm is developed for solving clustering problems with the similarity measure defined using the L1and L∞ norms. It is based on an incremental approach and applies nonsmooth optimization methods to find cluster centers. Computational results on 12 data sets are reported and the proposed algorithm is compared with the X-means algorithm. ©

Derivative free algorithms for nonsmooth and global optimization with application in cluster analysis

**Authors:**Ganjehlou, Asef Nazari**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.**Description:**Doctor of Philosophy

**Authors:**Ganjehlou, Asef Nazari**Date:**2009**Type:**Text , Thesis , PhD**Full Text:****Description:**This thesis is devoted to the development of algorithms for solving nonsmooth nonconvex problems. Some of these algorithms are derivative free methods.**Description:**Doctor of Philosophy

An algorithm for clustering using L1-norm based on hyperbolic smoothing technique

- Bagirov, Adil, Mohebi, Ehsan

**Authors:**Bagirov, Adil , Mohebi, Ehsan**Date:**2016**Type:**Text , Journal article**Relation:**Computational Intelligence Vol. 32, no. 3 (2016), p. 439-457**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Cluster analysis deals with the problem of organization of a collection of objects into clusters based on a similarity measure, which can be defined using various distance functions. The use of different similarity measures allows one to find different cluster structures in a data set. In this article, an algorithm is developed to solve clustering problems where the similarity measure is defined using the L1-norm. The algorithm is designed using the nonsmooth optimization approach to the clustering problem. Smoothing techniques are applied to smooth both the clustering function and the L1-norm. The algorithm computes clusters sequentially and finds global or near global solutions to the clustering problem. Results of numerical experiments using 12 real-world data sets are reported, and the proposed algorithm is compared with two other clustering algorithms. ©2015 Wiley Periodicals, Inc.

Clustering in large data sets with the limited memory bundle method

- Karmitsa, Napsu, Bagirov, Adil, Taheri, Sona

**Authors:**Karmitsa, Napsu , Bagirov, Adil , Taheri, Sona**Date:**2018**Type:**Text , Journal article**Relation:**Pattern Recognition Vol. 83, no. (2018), p. 245-259**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The aim of this paper is to design an algorithm based on nonsmooth optimization techniques to solve the minimum sum-of-squares clustering problems in very large data sets. First, the clustering problem is formulated as a nonsmooth optimization problem. Then the limited memory bundle method [Haarala et al., 2007] is modified and combined with an incremental approach to design a new clustering algorithm. The algorithm is evaluated using real world data sets with both the large number of attributes and the large number of data points. It is also compared with some other optimization based clustering algorithms. The numerical results demonstrate the efficiency of the proposed algorithm for clustering in very large data sets.

**Authors:**Bagirov, Adil , Ugon, Julien**Date:**2018**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 33, no. 1 (2018), p. 194-219**Full Text:**false**Reviewed:****Description:**The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets.

An incremental clustering algorithm based on hyperbolic smoothing

- Bagirov, Adil, Ordin, Burak, Ozturk, Gurkan, Xavier, Adilson

**Authors:**Bagirov, Adil , Ordin, Burak , Ozturk, Gurkan , Xavier, Adilson**Date:**2015**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 61, no. 1 (2015), p. 219-241**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clustering is an important problem in data mining. It can be formulated as a nonsmooth, nonconvex optimization problem. For the most global optimization techniques this problem is challenging even in medium size data sets. In this paper, we propose an approach that allows one to apply local methods of smooth optimization to solve the clustering problems. We apply an incremental approach to generate starting points for cluster centers which enables us to deal with nonconvexity of the problem. The hyperbolic smoothing technique is applied to handle nonsmoothness of the clustering problems and to make it possible application of smooth optimization algorithms to solve them. Results of numerical experiments with eleven real-world data sets and the comparison with state-of-the-art incremental clustering algorithms demonstrate that the smooth optimization algorithms in combination with the incremental approach are powerful alternative to existing clustering algorithms.

Nonsmooth optimization based algorithms in cluster analysis

- Bagirov, Adil, Mohebi, Ehsan

**Authors:**Bagirov, Adil , Mohebi, Ehsan**Date:**2015**Type:**Text , Book chapter**Relation:**Partitional Clustering Algorithms p. 99-146**Full Text:**false**Reviewed:****Description:**Cluster analysis is an important task in data mining. It deals with the problem of organization of a collection of objects into clusters based on a similarity measure. Various distance functions can be used to define the similarity measure. Cluster analysis problems with the similarity measure defined by the squared Euclidean distance, which is also known as the minimum sum-of-squares clustering, has been studied extensively over the last five decades. L1 and L1 norms have attracted less attention. In this chapter, we consider a nonsmooth nonconvex optimization formulation of the cluster analysis problems. This formulation allows one to easily apply similarity measures defined using different distance functions. Moreover, an efficient incremental algorithm can be designed based on this formulation to solve the clustering problems. We develop incremental algorithms for solving clustering problems where the similarity measure is defined using the L1; L2 and L1 norms. We also consider different algorithms for solving nonsmooth nonconvex optimization problems in cluster analysis. The proposed algorithms are tested using several real world data sets and compared with other similar algorithms.**Description:**Cluster analysis is an important task in data mining. It deals with the problem of organization of a collection of objects into clusters based on a similarity measure. Various distance functions can be used to define the similarity measure. Cluster analysis problems with the similarity measure defined by the squared Euclidean distance, which is also known as the minimum sum-of-squares clustering, has been studied extensively over the last five decades. However, problems with the L

**Authors:**Bagirov, Adil , Ugon, Julien**Date:**2018**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 33, no. 1 (2018), p. 194-219**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem using the squared regression error function. The objective function in this problem is represented as a difference of convex functions. Optimality conditions are derived, and an algorithm is designed based on such a representation. An incremental approach is proposed to generate starting solutions. The algorithm is tested on small to large data sets. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

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