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20Bagirov, Adil
10Karmitsa, Napsu
10Mammadov, Musa
4Joki, Kaisa
4Mäkelä, Marko
2Brown, Simon
2Gaudioso, Manlio
2Gondal, Iqbal
2Seifollahi, Sattar
1Al Nuaimat, Alia
1Asadi, Soodabeh
1Bai, Fusheng
1Chi, Chihung
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11Nonsmooth optimization
80102 Applied Mathematics
70103 Numerical and Computational Mathematics
7Nonconvex optimization
5DC optimization
40801 Artificial Intelligence and Image Processing
4Bayesian networks
4Bundle methods
4Optimization
308 Information and Computing Sciences
30906 Electrical and Electronic Engineering
3Cluster analysis
3Data mining
3Incremental approach
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2Algorithms
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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.

A globally optimization algorithm for systems of nonlinear equations

- Mammadov, Musa, Taheri, Sona

**Authors:**Mammadov, Musa , Taheri, Sona**Date:**2010**Type:**Text , Conference proceedings**Full Text:**false**Description:**In this paper, a new algorithm is proposed for the solutions of system of nonlinear equations. This algorithm uses a combination of the gradient and Newton's methods. A novel dynamic combinator is developed to determine the contribution of the methods in the combination. Also, by using some parameters in the proposed algorithm, this contribution is adjusted. The efficiency of the algoritms is studied in solving system of nonlinear equations.

A globally optimization algorithm for systems of nonlinear equations

- Mammadov, Musa, Taheri, Sona

**Authors:**Mammadov, Musa , Taheri, Sona**Date:**2010**Type:**Text , Conference paper**Relation:**Proceedings of PCO 2010, The Third International Conference on Power Control and Optimization 2010 Gold Coast p. 214-234**Full Text:**false**Reviewed:****Description:**In this paper, a new algorithm is proposed for the solutions of system of nonlinear equations. This algorithm uses a combination of the gradient and Newton's methods. A novel dynamic combinator is developed to determine the contribution of the methods in the combination. Also, by using some parameters in the proposed algorithm, this contribution is adjusted. The efficiency of the algoritms is studied in solving system of nonlinear equations.

Aggregate subgradient method for nonsmooth DC optimization

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

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

- 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.

An augmented subgradient method for minimizing nonsmooth DC functions

- Bagirov, Adil, Hoseini Monjezi, Najmeh, Taheri, Sona

**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.

Attribute weighted Naive Bayes classifier using a local optimization

- Taheri, Sona, Yearwood, John, Mammadov, Musa, Seifollahi, Sattar

**Authors:**Taheri, Sona , Yearwood, John , Mammadov, Musa , Seifollahi, Sattar**Date:**2013**Type:**Text , Journal article**Relation:**Neural Computing & Applications Vol.24, no.5 (2013), p.995-1002**Full Text:****Reviewed:****Description:**The Naive Bayes classifier is a popular classification technique for data mining and machine learning. It has been shown to be very effective on a variety of data classification problems. However, the strong assumption that all attributes are conditionally independent given the class is often violated in real-world applications. Numerous methods have been proposed in order to improve the performance of the Naive Bayes classifier by alleviating the attribute independence assumption. However, violation of the independence assumption can increase the expected error. Another alternative is assigning the weights for attributes. In this paper, we propose a novel attribute weighted Naive Bayes classifier by considering weights to the conditional probabilities. An objective function is modeled and taken into account, which is based on the structure of the Naive Bayes classifier and the attribute weights. The optimal weights are determined by a local optimization method using the quasisecant method. In the proposed approach, the Naive Bayes classifier is taken as a starting point. We report the results of numerical experiments on several real-world data sets in binary classification, which show the efficiency of the proposed method.

**Authors:**Taheri, Sona , Yearwood, John , Mammadov, Musa , Seifollahi, Sattar**Date:**2013**Type:**Text , Journal article**Relation:**Neural Computing & Applications Vol.24, no.5 (2013), p.995-1002**Full Text:****Reviewed:****Description:**The Naive Bayes classifier is a popular classification technique for data mining and machine learning. It has been shown to be very effective on a variety of data classification problems. However, the strong assumption that all attributes are conditionally independent given the class is often violated in real-world applications. Numerous methods have been proposed in order to improve the performance of the Naive Bayes classifier by alleviating the attribute independence assumption. However, violation of the independence assumption can increase the expected error. Another alternative is assigning the weights for attributes. In this paper, we propose a novel attribute weighted Naive Bayes classifier by considering weights to the conditional probabilities. An objective function is modeled and taken into account, which is based on the structure of the Naive Bayes classifier and the attribute weights. The optimal weights are determined by a local optimization method using the quasisecant method. In the proposed approach, the Naive Bayes classifier is taken as a starting point. We report the results of numerical experiments on several real-world data sets in binary classification, which show the efficiency of the proposed 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.

Clusterwise support vector linear regression

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

**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:**false**Reviewed:****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

- Taheri, Sona, Bagirov, Adil, Gondal, Iqbal, Brown, Simon

**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**Full Text:****Reviewed:****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).

**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**Full Text:****Reviewed:****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).

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.

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

**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**Reviewed:****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.

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.

- Bagirov, Adil, Gaudioso, Manlio, Karmitsa, Napsu, Mäkelä, Marko, Taheri, Sona

**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:**

Globally convergent algorithms for solving unconstrained optimization problems

- Taheri, Sona, Mammadov, Musa, Seifollahi, Sattar

**Authors:**Taheri, Sona , Mammadov, Musa , Seifollahi, Sattar**Date:**2013**Type:**Text , Journal article**Relation:**Optimization Vol. , no. (2013), p. 1-15**Full Text:****Reviewed:****Description:**New algorithms for solving unconstrained optimization problems are presented based on the idea of combining two types of descent directions: the direction of anti-gradient and either the Newton or quasi-Newton directions. The use of latter directions allows one to improve the convergence rate. Global and superlinear convergence properties of these algorithms are established. Numerical experiments using some unconstrained test problems are reported. Also, the proposed algorithms are compared with some existing similar methods using results of experiments. This comparison demonstrates the efficiency of the proposed combined methods.

**Authors:**Taheri, Sona , Mammadov, Musa , Seifollahi, Sattar**Date:**2013**Type:**Text , Journal article**Relation:**Optimization Vol. , no. (2013), p. 1-15**Full Text:****Reviewed:****Description:**New algorithms for solving unconstrained optimization problems are presented based on the idea of combining two types of descent directions: the direction of anti-gradient and either the Newton or quasi-Newton directions. The use of latter directions allows one to improve the convergence rate. Global and superlinear convergence properties of these algorithms are established. Numerical experiments using some unconstrained test problems are reported. Also, the proposed algorithms are compared with some existing similar methods using results of experiments. This comparison demonstrates the efficiency of the proposed combined methods.

Improving Naive Bayes classifier using conditional probabilities

- Taheri, Sona, Mammadov, Musa, Bagirov, Adil

**Authors:**Taheri, Sona , Mammadov, Musa , Bagirov, Adil**Date:**2010**Type:**Text , Conference proceedings**Full Text:****Description:**Naive Bayes classifier is the simplest among Bayesian Network classifiers. It has shown to be very efficient on a variety of data classification problems. However, the strong assumption that all features are conditionally independent given the class is often violated on many real world applications. Therefore, improvement of the Naive Bayes classifier by alleviating the feature independence assumption has attracted much attention. In this paper, we develop a new version of the Naive Bayes classifier without assuming independence of features. The proposed algorithm approximates the interactions between features by using conditional probabilities. We present results of numerical experiments on several real world data sets, where continuous features are discretized by applying two different methods. These results demonstrate that the proposed algorithm significantly improve the performance of the Naive Bayes classifier, yet at the same time maintains its robustness. © 2011, Australian Computer Society, Inc.**Description:**2003009505

**Authors:**Taheri, Sona , Mammadov, Musa , Bagirov, Adil**Date:**2010**Type:**Text , Conference proceedings**Full Text:****Description:**Naive Bayes classifier is the simplest among Bayesian Network classifiers. It has shown to be very efficient on a variety of data classification problems. However, the strong assumption that all features are conditionally independent given the class is often violated on many real world applications. Therefore, improvement of the Naive Bayes classifier by alleviating the feature independence assumption has attracted much attention. In this paper, we develop a new version of the Naive Bayes classifier without assuming independence of features. The proposed algorithm approximates the interactions between features by using conditional probabilities. We present results of numerical experiments on several real world data sets, where continuous features are discretized by applying two different methods. These results demonstrate that the proposed algorithm significantly improve the performance of the Naive Bayes classifier, yet at the same time maintains its robustness. © 2011, Australian Computer Society, Inc.**Description:**2003009505

Incremental DC optimization algorithm for large-scale clusterwise linear regression

- Bagirov, Adil, Taheri, Sona, Cimen, Emre

**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.

- Bagirov, Adil, Gaudioso, Manlio, Karmitsa, Napsu, Mäkelä, Marko, Taheri, Sona

**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**Reviewed:****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.

Learning Bayesian networks based on optimization approaches

**Authors:**Taheri, Sona**Date:**2012**Type:**Text , Thesis , PhD**Full Text:****Description:**Learning accurate classifiers from preclassified data is a very active research topic in machine learning and artifcial intelligence. There are numerous classifier paradigms, among which Bayesian Networks are very effective and well known in domains with uncertainty. Bayesian Networks are widely used representation frameworks for reasoning with probabilistic information. These models use graphs to capture dependence and independence relationships between feature variables, allowing a concise representation of the knowledge as well as efficient graph based query processing algorithms. This representation is defined by two components: structure learning and parameter learning. The structure of this model represents a directed acyclic graph. The nodes in the graph correspond to the feature variables in the domain, and the arcs (edges) show the causal relationships between feature variables. A directed edge relates the variables so that the variable corresponding to the terminal node (child) will be conditioned on the variable corresponding to the initial node (parent). The parameter learning represents probabilities and conditional probabilities based on prior information or past experience. The set of probabilities are represented in the conditional probability table. Once the network structure is constructed, the probabilistic inferences are readily calculated, and can be performed to predict the outcome of some variables based on the observations of others. However, the problem of structure learning is a complex problem since the number of candidate structures grows exponentially when the number of feature variables increases. This thesis is devoted to the development of learning structures and parameters in Bayesian Networks. Different models based on optimization techniques are introduced to construct an optimal structure of a Bayesian Network. These models also consider the improvement of the Naive Bayes' structure by developing new algorithms to alleviate the independence assumptions. We present various models to learn parameters of Bayesian Networks; in particular we propose optimization models for the Naive Bayes and the Tree Augmented Naive Bayes by considering different objective functions. To solve corresponding optimization problems in Bayesian Networks, we develop new optimization algorithms. Local optimization methods are introduced based on the combination of the gradient and Newton methods. It is proved that the proposed methods are globally convergent and have superlinear convergence rates. As a global search we use the global optimization method, AGOP, implemented in the open software library GANSO. We apply the proposed local methods in the combination with AGOP. Therefore, the main contributions of this thesis include (a) new algorithms for learning an optimal structure of a Bayesian Network; (b) new models for learning the parameters of Bayesian Networks with the given structures; and finally (c) new optimization algorithms for optimizing the proposed models in (a) and (b). To validate the proposed methods, we conduct experiments across a number of real world problems. Print version is available at: http://library.federation.edu.au/record=b1804607~S4**Description:**Doctor of Philosophy

Learning the naive bayes classifier with optimization models

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2013**Type:**Text , Journal article**Relation:**International Journal of Applied Mathematics and Computer Science Vol. 23, no. 4 (2013), p. 787-795**Full Text:****Reviewed:****Description:**Naive Bayes is among the simplest probabilistic classifiers. It often performs surprisingly well in many real world applications, despite the strong assumption that all features are conditionally independent given the class. In the learning process of this classifier with the known structure, class probabilities and conditional probabilities are calculated using training data, and then values of these probabilities are used to classify new observations. In this paper, we introduce three novel optimization models for the naive Bayes classifier where both class probabilities and conditional probabilities are considered as variables. The values of these variables are found by solving the corresponding optimization problems. Numerical experiments are conducted on several real world binary classification data sets, where continuous features are discretized by applying three different methods. The performances of these models are compared with the naive Bayes classifier, tree augmented naive Bayes, the SVM, C4.5 and the nearest neighbor classifier. The obtained results demonstrate that the proposed models can significantly improve the performance of the naive Bayes classifier, yet at the same time maintain its simple structure.

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2013**Type:**Text , Journal article**Relation:**International Journal of Applied Mathematics and Computer Science Vol. 23, no. 4 (2013), p. 787-795**Full Text:****Reviewed:****Description:**Naive Bayes is among the simplest probabilistic classifiers. It often performs surprisingly well in many real world applications, despite the strong assumption that all features are conditionally independent given the class. In the learning process of this classifier with the known structure, class probabilities and conditional probabilities are calculated using training data, and then values of these probabilities are used to classify new observations. In this paper, we introduce three novel optimization models for the naive Bayes classifier where both class probabilities and conditional probabilities are considered as variables. The values of these variables are found by solving the corresponding optimization problems. Numerical experiments are conducted on several real world binary classification data sets, where continuous features are discretized by applying three different methods. The performances of these models are compared with the naive Bayes classifier, tree augmented naive Bayes, the SVM, C4.5 and the nearest neighbor classifier. The obtained results demonstrate that the proposed models can significantly improve the performance of the naive Bayes classifier, yet at the same time maintain its simple structure.

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