A novel optimization approach towards improving separability of clusters
- Authors: Bagirov, Adil , Hoseini-Monjezi, Najmeh , Taheri, Sona
- Date: 2023
- Type: Text , Journal article
- Relation: Computers and Operations Research Vol. 152, no. (2023), p.
- Relation: http://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
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- Description: The objective functions in optimization models of the sum-of-squares clustering problem reflect intra-cluster similarity and inter-cluster dissimilarities and in general, optimal values of these functions can be considered as appropriate measures for compactness of clusters. However, the use of the objective function alone may not lead to the finding of separable clusters. To address this shortcoming in existing models for clustering, we develop a new optimization model where the objective function is represented as a sum of two terms reflecting the compactness and separability of clusters. Based on this model we develop a two-phase incremental clustering algorithm. In the first phase, the clustering function is minimized to find compact clusters and in the second phase, a new model is applied to improve the separability of clusters. The Davies–Bouldin cluster validity index is applied as an additional measure to compare the compactness of clusters and silhouette coefficients are used to estimate the separability of clusters. The performance of the proposed algorithm is demonstrated and compared with that of four other algorithms using synthetic and real-world data sets. Numerical results clearly show that in comparison with other algorithms the new algorithm is able to find clusters with better separability and similar compactness. © 2022
Bundle enrichment method for nonsmooth difference of convex programming problems
- Authors: Gaudioso, Manilo , Taheri, Sona , Bagirov, Adil , Karmitsa, Napsu
- Date: 2023
- Type: Text , Journal article
- Relation: Algorithms Vol. 16, no. 8 (2023), p.
- Relation: http://purl.org/au-research/grants/arc/DP190100580
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- Description: The Bundle Enrichment Method (BEM-DC) is introduced for solving nonsmooth difference of convex (DC) programming problems. The novelty of the method consists of the dynamic management of the bundle. More specifically, a DC model, being the difference of two convex piecewise affine functions, is formulated. The (global) minimization of the model is tackled by solving a set of convex problems whose cardinality depends on the number of linearizations adopted to approximate the second DC component function. The new bundle management policy distributes the information coming from previous iterations to separately model the DC components of the objective function. Such a distribution is driven by the sign of linearization errors. If the displacement suggested by the model minimization provides no sufficient decrease of the objective function, then the temporary enrichment of the cutting plane approximation of just the first DC component function takes place until either the termination of the algorithm is certified or a sufficient decrease is achieved. The convergence of the BEM-DC method is studied, and computational results on a set of academic test problems with nonsmooth DC objective functions are provided. © 2023 by the authors.
Methods and applications of clusterwise linear regression : a survey and comparison
- Authors: Long, Qiang , Bagirov, Adil , Taheri, Sona , Sultanova, Nargiz , Wu, Xue
- Date: 2023
- Type: Text , Journal article
- Relation: ACM Transactions on Knowledge Discovery from Data Vol. 17, no. 3 (2023), p.
- Relation: https://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
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- Description: Clusterwise linear regression (CLR) is a well-known technique for approximating a data using more than one linear function. It is based on the combination of clustering and multiple linear regression methods. This article provides a comprehensive survey and comparative assessments of CLR including model formulations, description of algorithms, and their performance on small to large-scale synthetic and real-world datasets. Some applications of the CLR algorithms and possible future research directions are also discussed. © 2023 Association for Computing Machinery.
Nonsmooth optimization-based hyperparameter-free neural networks for large-scale regression
- Authors: Karmitsa, Napsu , Taheri, Sona , Joki, Kaisa , Paasivirta, Pauliina , Defterdarovic, J. , Bagirov, Adil , Mäkelä, Marko
- Date: 2023
- Type: Text , Journal article
- Relation: Algorithms Vol. 16, no. 9 (2023), p.
- Relation: http://purl.org/au-research/grants/arc/DP190100580
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- Description: In this paper, a new nonsmooth optimization-based algorithm for solving large-scale regression problems is introduced. The regression problem is modeled as fully-connected feedforward neural networks with one hidden layer, piecewise linear activation, and the (Formula presented.) -loss functions. A modified version of the limited memory bundle method is applied to minimize this nonsmooth objective. In addition, a novel constructive approach for automated determination of the proper number of hidden nodes is developed. Finally, large real-world data sets are used to evaluate the proposed algorithm and to compare it with some state-of-the-art neural network algorithms for regression. The results demonstrate the superiority of the proposed algorithm as a predictive tool in most data sets used in numerical experiments. © 2023 by the authors.
Nonsmooth optimization-based model and algorithm for semisupervised clustering
- Authors: Bagirov, Adil , Taheri, Sona , Bai, Fusheng , Zheng, Fangying
- Date: 2023
- Type: Text , Journal article
- Relation: IEEE Transactions on Neural Networks and Learning Systems Vol. 34, no. 9 (2023), p. 5517-5530
- Relation: http://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
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- Description: Using a nonconvex nonsmooth optimization approach, we introduce a model for semisupervised clustering (SSC) with pairwise constraints. In this model, the objective function is represented as a sum of three terms: the first term reflects the clustering error for unlabeled data points, the second term expresses the error for data points with must-link (ML) constraints, and the third term represents the error for data points with cannot-link (CL) constraints. This function is nonconvex and nonsmooth. To find its optimal solutions, we introduce an adaptive SSC (A-SSC) algorithm. This algorithm is based on the combination of the nonsmooth optimization method and an incremental approach, which involves the auxiliary SSC problem. The algorithm constructs clusters incrementally starting from one cluster and gradually adding one cluster center at each iteration. The solutions to the auxiliary SSC problem are utilized as starting points for solving the nonconvex SSC problem. The discrete gradient method (DGM) of nonsmooth optimization is applied to solve the underlying nonsmooth optimization problems. This method does not require subgradient evaluations and uses only function values. The performance of the A-SSC algorithm is evaluated and compared with four benchmarking SSC algorithms on one synthetic and 12 real-world datasets. Results demonstrate that the proposed algorithm outperforms the other four algorithms in identifying compact and well-separated clusters while satisfying most constraints. © 2021 IEEE.
Limited Memory Bundle Method for Clusterwise Linear Regression
- Authors: Karmitsa, Napsu , Bagirov, Adil , Taheri, Sona , Joki, Kaisa
- Date: 2022
- Type: Text , Book chapter
- Relation: Intelligent Systems, Control and Automation: Science and Engineering p. 109-122
- Relation: http://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
- Reviewed:
- Description: A clusterwise linear regression problem consists of finding a number of linear functions each approximating a subset of the given data. In this paper, the limited memory bundle method is modified and combined with the incremental approach to solve this problem using its nonsmooth optimization formulation. The main contribution of the proposed method is to obtain a fast solution time for large-scale clusterwise linear regression problems. The proposed algorithm is tested on small and large real-world data sets and compared with other algorithms for clusterwise linear regression. Numerical results demonstrate that the proposed algorithm is especially efficient in data sets with large numbers of data points and input variables. © 2022, Springer Nature Switzerland AG.
Missing value imputation via clusterwise linear regression
- Authors: Karmitsa, Napsu , Taheri, Sona , Bagirov, Adil , Makinen, Pauliina
- Date: 2022
- Type: Text , Journal article
- Relation: IEEE transactions on knowledge and data engineering Vol. 34, no. 4 (2020), p. 1889-1901
- Full Text: false
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- Description:
In this paper a new method of preprocessing incomplete data is introduced. The method is based on clusterwise linear regression and it combines two well-known approaches for missing value imputation: linear regression and clustering. The idea is to approximate missing values using only those data points that are somewhat similar to the incomplete data point. A similar idea is used also in clustering based imputation methods. Nevertheless, here the linear regression approach is used within each cluster to accurately predict the missing values, and this is done simultaneously to clustering. The proposed method is tested using some synthetic and real-world data sets and compared with other algorithms for missing value imputations. Numerical results demonstrate that the proposed method produces the most accurate imputations in MCAR and MAR data sets with a clear structure and the percentages of missing data no more than 25%
Robust piecewise linear L 1-regression via nonsmooth DC optimization
- Authors: Bagirov, Adil , Taheri, Sona , Karmitsa, Napsu , Sultanova, Nargiz , Asadi, Soodabeh
- Date: 2022
- Type: Text , Journal article
- Relation: Optimization Methods and Software Vol. 37, no. 4 (2022), p. 1289-1309
- Relation: http://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
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- Description: Piecewise linear (Formula presented.) -regression problem is formulated as an unconstrained difference of convex (DC) optimization problem and an algorithm for solving this problem is developed. Auxiliary problems are introduced to design an adaptive approach to generate a suitable piecewise linear regression model and starting points for solving the underlying DC optimization problems. The performance of the proposed algorithm as both approximation and prediction tool is evaluated using synthetic and real-world data sets containing outliers. It is also compared with mainstream machine learning regression algorithms using various performance measures. Results demonstrate that the new algorithm is robust to outliers and in general, provides better predictions than the other alternative regression algorithms for most data sets used in the numerical experiments. © 2020 Informa UK Limited, trading as Taylor & Francis Group.
Aggregate subgradient method for nonsmooth DC optimization
- Authors: Bagirov, Adil , Taheri, Sona , Joki, Kaisa , Karmitsa, Napsu , Mäkelä, Marko
- Date: 2021
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 15, no. 1 (2021), p. 83-96
- Relation: http://purl.org/au-research/grants/arc/DP190100580
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- Description: The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
An augmented subgradient method for minimizing nonsmooth DC functions
- Authors: Bagirov, Adil , Hoseini Monjezi, Najmeh , Taheri, Sona
- Date: 2021
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 80, no. 2 (2021), p. 411-438
- Relation: http://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
- Reviewed:
- Description: A method, called an augmented subgradient method, is developed to solve unconstrained nonsmooth difference of convex (DC) optimization problems. At each iteration of this method search directions are found by using several subgradients of the first DC component and one subgradient of the second DC component of the objective function. The developed method applies an Armijo-type line search procedure to find the next iteration point. It is proved that the sequence of points generated by the method converges to a critical point of the unconstrained DC optimization problem. The performance of the method is demonstrated using academic test problems with nonsmooth DC objective functions and its performance is compared with that of two general nonsmooth optimization solvers and five solvers specifically designed for unconstrained DC optimization. Computational results show that the developed method is efficient and robust for solving nonsmooth DC optimization problems. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Incremental DC optimization algorithm for large-scale clusterwise linear regression
- Authors: Bagirov, Adil , Taheri, Sona , Cimen, Emre
- Date: 2021
- Type: Text , Journal article
- Relation: Journal of Computational and Applied Mathematics Vol. 389, no. (2021), p. 1-17
- Relation: https://purl.org/au-research/grants/arc/DP190100580
- Full Text: false
- Reviewed:
- Description: The objective function in the nonsmooth optimization model of the clusterwise linear regression (CLR) problem with the squared regression error is represented as a difference of two convex functions. Then using the difference of convex algorithm (DCA) approach the CLR problem is replaced by the sequence of smooth unconstrained optimization subproblems. A new algorithm based on the DCA and the incremental approach is designed to solve the CLR problem. We apply the Quasi-Newton method to solve the subproblems. The proposed algorithm is evaluated using several synthetic and real-world data sets for regression and compared with other algorithms for CLR. Results demonstrate that the DCA based algorithm is efficient for solving CLR problems with the large number of data points and in particular, outperforms other algorithms when the number of input variables is small. © 2020 Elsevier B.V.
Subgradient smoothing method for nonsmooth nonconvex optimization
- Authors: Bagirov, Adil , Sultanova, Nargiz , Taheri, Sona , Ozturk, Gurkan
- Date: 2021
- Type: Text , Conference paper
- Relation: 5th International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOV, Muscan, 6-9 January 2020 Vol. 354, p. 57-79
- Full Text: false
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- Description: In this chapter an unconstrained nonsmooth nonconvex optimization problem is considered and a method for solving this problem is developed. In this method the subproblem for finding search directions is reduced to the unconstrained minimization of a smooth function. This is achieved by using subgradients computed in some neighborhood of a current iteration point and by formulating the search direction finding problem to the minimization of the convex piecewise linear function over the unit ball. The hyperbolic smoothing technique is applied to approximate the minimization problem by a sequence of smooth problems. The convergence of the proposed method is studied and its performance is evaluated using a set of nonsmooth optimization academic test problems. In addition, the method is implemented in GAMS and numerical results using different solvers from GAMS are reported. The proposed method is compared with a number of nonsmooth optimization methods. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Clusterwise support vector linear regression
- Authors: Joki, Kaisa , Bagirov, Adil , Karmitsa, Napsu , Mäkelä, Marko , Taheri, Sona
- Date: 2020
- Type: Text , Journal article
- Relation: European Journal of Operational Research Vol. 287, no. 1 (2020), p. 19-35
- Full Text:
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- Description: In clusterwise linear regression (CLR), the aim is to simultaneously partition data into a given number of clusters and to find regression coefficients for each cluster. In this paper, we propose a novel approach to model and solve the CLR problem. The main idea is to utilize the support vector machine (SVM) approach to model the CLR problem by using the SVM for regression to approximate each cluster. This new formulation of the CLR problem is represented as an unconstrained nonsmooth optimization problem, where we minimize a difference of two convex (DC) functions. To solve this problem, a method based on the combination of the incremental algorithm and the double bundle method for DC optimization is designed. Numerical experiments are performed to validate the reliability of the new formulation for CLR and the efficiency of the proposed method. The results show that the SVM approach is suitable for solving CLR problems, especially, when there are outliers in data. © 2020 Elsevier B.V.
- Description: Funding details: Academy of Finland, 289500, 294002, 319274 Funding details: Turun Yliopisto Funding details: Australian Research Council, ARC, (Project no. DP190100580 ).
Cyberattack triage using incremental clustering for intrusion detection systems
- Authors: Taheri, Sona , Bagirov, Adil , Gondal, Iqbal , Brown, Simon
- Date: 2020
- Type: Text , Journal article
- Relation: International Journal of Information Security Vol. 19, no. 5 (2020), p. 597-607
- Relation: http://purl.org/au-research/grants/arc/DP190100580
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- Description: Intrusion detection systems (IDSs) are devices or software applications that monitor networks or systems for malicious activities and signals alerts/alarms when such activity is discovered. However, an IDS may generate many false alerts which affect its accuracy. In this paper, we develop a cyberattack triage algorithm to detect these alerts (so-called outliers). The proposed algorithm is designed using the clustering, optimization and distance-based approaches. An optimization-based incremental clustering algorithm is proposed to find clusters of different types of cyberattacks. Using a special procedure, a set of clusters is divided into two subsets: normal and stable clusters. Then, outliers are found among stable clusters using an average distance between centroids of normal clusters. The proposed algorithm is evaluated using the well-known IDS data sets—Knowledge Discovery and Data mining Cup 1999 and UNSW-NB15—and compared with some other existing algorithms. Results show that the proposed algorithm has a high detection accuracy and its false negative rate is very low. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
- Description: This research was conducted in Internet Commerce Security Laboratory (ICSL) funded by Westpac Banking Corporation Australia. In addition, the research by Dr. Sona Taheri and A/Prof. Adil Bagirov was supported by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (DP190100580).
Discrete gradient methods
- Authors: Bagirov, Adil , Taheri, Sona , Karmitsa, Napsu
- Date: 2020
- Type: Text , Book chapter
- Relation: Numerical Nonsmooth Optimization: State of the Art Algorithms p. 621-654
- Full Text: false
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- Description: In this chapter, the notion of a discrete gradient is introduced and it is shown that the discrete gradients can be used to approximate subdifferentials of a broad class of nonsmooth functions. Two methods based on such approximations, more specifically, the discrete gradient method (DGM) and its limited memory version (LDGB), are described. These methods are semi derivative-free methods for solving nonsmooth and, in general, nonconvex optimization problems. The performance of the methods is demonstrated using some academic test problems. © Springer Nature Switzerland AG 2020.
Final words
- Authors: Bagirov, Adil , Gaudioso, Manlio , Karmitsa, Napsu , Mäkelä, Marko , Taheri, Sona
- Date: 2020
- Type: Text , Book chapter
- Relation: Numerical Nonsmooth Optimization: State of the Art Algorithms p. 693-694
- Full Text: false
- Reviewed:
Introduction
- Authors: Bagirov, Adil , Gaudioso, Manlio , Karmitsa, Napsu , Mäkelä, Marko , Taheri, Sona
- Date: 2020
- Type: Text , Book chapter
- Relation: Numerical Nonsmooth Optimization: State of the Art Algorithms p. 1-16
- Full Text: false
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- Description: Nonsmooth optimization (NSO) is among the most challenging tasks in the field of mathematical programming. It addresses optimization problems where objective and/or constraint functions have discontinuous gradients. NSO problems arise in many real life applications. Moreover, some smooth optimization techniques like different decomposition methods, dual formulations and exact penalty methods may lead us to solve NSO problems being either smaller in dimension or simpler in structure. In addition, some optimization problems may be analytically smooth but numerically nonsmooth. This is the case, for instance, with noisy input data and so-called stiff problems, which are numerically unstable and behave like nonsmooth problems. © Springer Nature Switzerland AG 2020.
A difference of convex optimization algorithm for piecewise linear regression
- 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.
An approximate ADMM for solving linearly constrained nonsmooth optimization problems with two blocks of variables
- 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.
Multi-source cyber-attacks detection using machine learning
- Authors: Taheri, Sona , Gondal, Iqbal , Bagirov, Adil , Harkness, Greg , Brown, Simon , Chi, Chihung
- Date: 2019
- Type: Text , Conference proceedings , Conference paper
- Relation: 2019 IEEE International Conference on Industrial Technology, ICIT 2019; Melbourne, Australia; 13th-15th February 2019 Vol. 2019-February, p. 1167-1172
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- Description: The Internet of Things (IoT) has significantly increased the number of devices connected to the Internet ranging from sensors to multi-source data information. As the IoT continues to evolve with new technologies number of threats and attacks against IoT devices are on the increase. Analyzing and detecting these attacks originating from different sources needs machine learning models. These models provide proactive solutions for detecting attacks and their sources. In this paper, we propose to apply a supervised machine learning classification technique to identify cyber-attacks from each source. More precisely, we apply the incremental piecewise linear classifier that constructs boundary between sources/classes incrementally starting with one hyperplane and adding more hyperplanes at each iteration. The algorithm terminates when no further significant improvement of the separation of sources/classes is possible. The construction and usage of piecewise linear boundaries allows us to avoid any possible overfitting. We apply the incremental piecewise linear classifier on the multi-source real world cyber security data set to identify cyber-attacks and their sources.
- Description: Proceedings of the IEEE International Conference on Industrial Technology