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11Ugon, Julien
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A difference of convex optimization algorithm for piecewise linear regression

- Bagirov, Adil, Taheri, Sona, Asadi, Soodabeh

**Authors:**Bagirov, Adil , Taheri, Sona , Asadi, Soodabeh**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 15, no. 2 (2019), p. 909-932**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The problem of finding a continuous piecewise linear function approximating a regression function is considered. This problem is formulated as a nonconvex nonsmooth optimization problem where the objective function is represented as a difference of convex (DC) functions. Subdifferentials of DC components are computed and an algorithm is designed based on these subdifferentials to find piecewise linear functions. The algorithm is tested using some synthetic and real world data sets and compared with other regression algorithms.

- Bagirov, Adil, Taheri, Sona, Bai, Fusheng, Wu, Zhiyou

**Authors:**Bagirov, Adil , Taheri, Sona , Bai, Fusheng , Wu, Zhiyou**Date:**2019**Type:**Text , Book chapter**Relation:**Nonsmooth Optimization and Its Applications (part of the International Series of Numerical Mathematics book series) Chapter 2 p. 17-44**Full Text:**false**Reviewed:****Description:**Nonsmooth convex optimization problems with two blocks of variables subject to linear constraints are considered. A new version of the alternating direction method of multipliers is developed for solving these problems. In this method the subproblems are solved approximately. The convergence of the method is studied. New test problems are designed and used to verify the efficiency of the proposed method and to compare it with two versions of the proximal bundle method.

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.

Double bundle method for finding clarke stationary points in nonsmooth dc programming

- Joki, Kaisa, Bagirov, Adil, Karmitsa, Napsu, Makela, Marko, Taheri, Sona

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko , Taheri, Sona**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 28, no. 2 (2018), p. 1892-1919**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:****Reviewed:****Description:**The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko , Taheri, Sona**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 28, no. 2 (2018), p. 1892-1919**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:****Reviewed:****Description:**The aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization, where the objective function is presented as a difference of two convex (DC) functions. The novelty in our method is a new escape procedure which enables us to guarantee approximate Clarke stationarity for solutions by utilizing the DC components of the objective function. This optimality condition is stronger than the criticality condition typically used in DC programming. Moreover, if a candidate solution is not approximate Clarke stationary, then the escape procedure returns a descent direction. With this escape procedure, we can avoid some shortcomings encountered when criticality is used. The finite termination of the double bundle method to an approximate Clarke stationary point is proved by assuming that the subdifferentials of DC components are polytopes. Finally, some encouraging numerical results are presented.

**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.

**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.

Solving minimax problems : Local smoothing versus global smoothing

- Bagirov, Adil, Sultanova, Nargiz, Al Nuaimat, Alia, Taheri, Sona

**Authors:**Bagirov, Adil , Sultanova, Nargiz , Al Nuaimat, Alia , Taheri, Sona**Date:**2018**Type:**Text , Conference proceedings**Relation:**4th International Conference on Numerical Analysis and Optimization, NAO-IV 2017; Muscat, Oman; 2nd-5th January 2017; published in Numerical Analysis and Optimization NAO-IV (part of the Springer Proceedings in Mathematics and Statistics book series PROMS, volume 235) Vol. 235, p. 23-43**Full Text:**false**Reviewed:****Description:**The aim of this chapter is to compare different smoothing techniques for solving finite minimax problems. We consider the local smoothing technique which approximates the function in some neighborhood of a point of nondifferentiability and also global smoothing techniques such as the exponential and hyperbolic smoothing which approximate the function in the whole domain. Computational results on the collection of academic test problems are used to compare different smoothing techniques. Results show the superiority of the local smoothing technique for convex problems and global smoothing techniques for nonconvex problems. © 2018, Springer International Publishing AG, part of Springer Nature.**Description:**Springer Proceedings in Mathematics and Statistics

A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes

- Joki, Kaisa, Bagirov, Adil, Karmitsa, Napsu, Makela, Marko

**Authors:**Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Makela, Marko**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 68, no. 3 (2017), p. 501-535**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an -critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.

DC programming algorithm for clusterwise linear L1 regression

**Authors:**Bagirov, Adil , Taheri, Sona**Date:**2017**Type:**Text , Journal article**Relation:**Journal of the Operations Research Society of China Vol. 5, no. 2 (2017), p. 233-256**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The aim of this paper is to develop an algorithm for solving the clusterwise linear least absolute deviations regression problem. This problem is formulated as a nonsmooth nonconvex optimization problem, and the objective function is represented as a difference of convex functions. Optimality conditions are derived by using this representation. An algorithm is designed based on the difference of convex representation and an incremental approach. The proposed algorithm is tested using small to large artificial and real-world data sets. © 2017, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg.

New diagonal bundle method for clustering problems in large data sets

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

**Authors:**Karmitsa, Napsu , Bagirov, Adil , Taheri, Sona**Date:**2017**Type:**Text , Journal article**Relation:**European Journal of Operational Research Vol. 263, no. 2 (2017), p. 367-379**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clustering is one of the most important tasks in data mining. Recent developments in computer hardware allow us to store in random access memory (RAM) and repeatedly read data sets with hundreds of thousands and even millions of data points. This makes it possible to use conventional clustering algorithms in such data sets. However, these algorithms may need prohibitively large computational time and fail to produce accurate solutions. Therefore, it is important to develop clustering algorithms which are accurate and can provide real time clustering in large data sets. This paper introduces one of them. Using nonsmooth optimization formulation of the clustering problem the objective function is represented as a difference of two convex (DC) functions. Then a new diagonal bundle algorithm that explicitly uses this structure is designed and combined with an incremental approach to solve this problem. The method is evaluated using real world data sets with both large number of attributes and large number of data points. The proposed method is compared with two other clustering algorithms using numerical results. © 2017 Elsevier B.V.

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.

A heuristic algorithm for solving the minimum sum-of-squares clustering problems

**Authors:**Ordin, Burak , Bagirov, Adil**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 61, no. 2 (2015), p. 341-361**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.

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.

An incremental piecewise linear classifier based on polyhedral conic separation

- Ozturk, Gurkan, Bagirov, Adil, Kasimbeyli, Refail

**Authors:**Ozturk, Gurkan , Bagirov, Adil , Kasimbeyli, Refail**Date:**2015**Type:**Text , Journal article**Relation:**Machine Learning Vol. 101, no. 1-3 (2015), p. 397-413**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**In this paper, a piecewise linear classifier based on polyhedral conic separation is developed. This classifier builds nonlinear boundaries between classes using polyhedral conic functions. Since the number of polyhedral conic functions separating classes is not known a priori, an incremental approach is proposed to build separating functions. These functions are found by minimizing an error function which is nonsmooth and nonconvex. A special procedure is proposed to generate starting points to minimize the error function and this procedure is based on the incremental approach. The discrete gradient method, which is a derivative-free method for nonsmooth optimization, is applied to minimize the error function starting from those points. The proposed classifier is applied to solve classification problems on 12 publicly available data sets and compared with some mainstream and piecewise linear classifiers. © 2014, The Author(s).

Nonsmooth optimization algorithm for solving clusterwise linear regression problems

- Bagirov, Adil, Ugon, Julien, Mirzayeva, Hijran

**Authors:**Bagirov, Adil , Ugon, Julien , Mirzayeva, Hijran**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 164, no. 3 (2015), p. 755-780**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**Clusterwise linear regression consists of finding a number of linear regression functions each approximating a subset of the data. In this paper, the clusterwise linear regression problem is formulated as a nonsmooth nonconvex optimization problem and an algorithm based on an incremental approach and on the discrete gradient method of nonsmooth optimization is designed to solve it. This algorithm incrementally divides the whole dataset into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate good starting points for solving global optimization problems at each iteration of the incremental algorithm. The algorithm is compared with the multi-start Spath and the incremental algorithms on several publicly available datasets for regression analysis.

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

Aggregate codifferential method for nonsmooth DC optimization

- Tor, Ali, Bagirov, Adil, Karasozen, Bulent

**Authors:**Tor, Ali , Bagirov, Adil , Karasozen, Bulent**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Computational and Applied Mathematics Vol. 259, no. Part B (2014), p. 851-867**Full Text:**false**Reviewed:****Description:**A new algorithm is developed based on the concept of codifferential for minimizing the difference of convex nonsmooth functions. Since the computation of the whole codifferential is not always possible, we use a fixed number of elements from the codifferential to compute the search directions. The convergence of the proposed algorithm is proved. The efficiency of the algorithm is demonstrated by comparing it with the subgradient, the truncated codifferential and the proximal bundle methods using nonsmooth optimization test problems.

An algorithm for clusterwise linear regression based on smoothing techniques

- Bagirov, Adil, Ugon, Julien, Mirzayeva, Hijran

**Authors:**Bagirov, Adil , Ugon, Julien , Mirzayeva, Hijran**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 2 (2014), p. 375-390**Full Text:**false**Reviewed:****Description:**We propose an algorithm based on an incremental approach and smoothing techniques to solve clusterwise linear regression (CLR) problems. This algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate an initial solution for solving global optimization problems at each iteration of the incremental algorithm. Such an approach allows one to find global or approximate global solutions to the CLR problems. The algorithm is tested using several data sets for regression analysis and compared with the multistart and incremental Spath algorithms.

Introduction to Nonsmooth Optimization : Theory, practice and software

- Bagirov, Adil, Karmitsa, Napsu, Makela, Marko

**Authors:**Bagirov, Adil , Karmitsa, Napsu , Makela, Marko**Date:**2014**Type:**Text , Book**Full Text:**false**Reviewed:****Description:**This book is the first easy-to-read text on nonsmooth optimization (NSO, not necessarily differentiable optimization). Soving these kinds of problems plays a critical role in many industrial applications and real-world modeling systems, for example in the context of image denoising, optimal control, neural network training, data mining, ecomonics, and computational chemistry and physics. The book covers both the theory and the numerical methods used in NSO, and provides an overview of different problems arising in the field. It is organized into three parts: 1. convex and nonconvex analysis and the theory of NSO; 2. test problems and practical applications; 3. a guide to NSO software. The book is ideal for anyone teaching or attending NSO courses. As an accessible introduction to the field, it is also well suited as an independent learning guide for practitioners already familiar with the basics of optimization.

Piecewise linear classifiers based on nonsmooth optimization approaches

- Bagirov, Adil, Kasimbeyli, Refail, Ozturk, Gurkan, Ugon, Julien

**Authors:**Bagirov, Adil , Kasimbeyli, Refail , Ozturk, Gurkan , Ugon, Julien**Date:**2014**Type:**Text , Book chapter**Relation:**Optimization in Science and Engineering p. 1-32**Full Text:**false**Reviewed:****Description:**Nonsmooth optimization provides efficient algorithms for solving many machine learning problems. In particular, nonsmooth optimization approaches to supervised data classification problems lead to the design of very efficient algorithms for their solution. In this chapter, we demonstrate how nonsmooth optimization algorithms can be applied to design efficient piecewise linear classifiers for supervised data classification problems. Such classifiers are developed using a max–min and a polyhedral conic separabilities as well as an incremental approach. We report results of numerical experiments and compare the piecewise linear classifiers with a number of other mainstream classifiers.

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