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5Nonconvex optimization
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2Regression analysis
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10801 Artificial Intelligence and Image Processing
10906 Electrical and Electronic Engineering
1Algorithms
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1Bayesian Networks
1Bundle methods
1Clarke stationarity
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1Clusterwise linear regression
1Clusterwise linear regressions
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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.

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.

Structure learning of Bayesian Networks using global optimization with applications in data classification

- Taheri, Sona, Mammadov, Musa

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

**Authors:**Taheri, Sona , Mammadov, Musa**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 9, no. 5 (2014), p. 931-948**Full Text:****Reviewed:****Description:**Bayesian Networks are increasingly popular methods of modeling uncertainty in artificial intelligence and machine learning. A Bayesian Network consists of a directed acyclic graph in which each node represents a variable and each arc represents probabilistic dependency between two variables. Constructing a Bayesian Network from data is a learning process that consists of two steps: learning structure and learning parameter. Learning a network structure from data is the most difficult task in this process. This paper presents a new algorithm for constructing an optimal structure for Bayesian Networks based on optimization. The algorithm has two major parts. First, we define an optimization model to find the better network graphs. Then, we apply an optimization approach for removing possible cycles from the directed graphs obtained in the first part which is the first of its kind in the literature. The main advantage of the proposed method is that the maximal number of parents for variables is not fixed a priory and it is defined during the optimization procedure. It also considers all networks including cyclic ones and then choose a best structure by applying a global optimization method. To show the efficiency of the algorithm, several closely related algorithms including unrestricted dependency Bayesian Network algorithm, as well as, benchmarks algorithms SVM and C4.5 are employed for comparison. We apply these algorithms on data classification; data sets are taken from the UCI machine learning repository and the LIBSVM. © 2014, Springer-Verlag Berlin Heidelberg.

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.

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.

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.

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.

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