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5Ugon, Julien
3Karmitsa, Napsu
2Joki, Kaisa
2Karasozen, Bulent
2Makela, Marko
2Mirzayeva, Hijran
2Ordin, Burak
2Sultanova, Nargiz
2Taheri, Sona
1Al Nuaimat, Alia
1Al Nuaimat, A.
1Asadi, Soodabeh
1Ganjehlou, Asef Nazari
1Jin, L.
1Ozturk, Gurkan
1Sezer, Monsalve
1Tor, Ali
1Xavier, Adilson

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140102 Applied Mathematics
9Nonconvex optimization
50802 Computation Theory and Mathematics
5Subdifferential
30906 Electrical and Electronic Engineering
3Cluster analysis
3DC functions
3DC programming
3Regression analysis
3Smoothing techniques
2Bundle method
2Codifferential
2Cutting plane model
10801 Artificial Intelligence and Image Processing
1Algorithm for solving
1Algorithms
1Bundle methods
1Clarke stationarity

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Max-min separability

**Authors:**Bagirov, Adil**Date:**2005**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 20, no. 2-3 (2005), p. 271-290**Full Text:****Reviewed:****Description:**We consider the problem of discriminating two finite point sets in the n-dimensional space by a finite number of hyperplanes generating a piecewise linear function. If the intersection of these sets is empty, then they can be strictly separated by a max-min of linear functions. An error function is introduced. This function is nonconvex piecewise linear. We discuss an algorithm for its minimization. The results of numerical experiments using some real-world datasets are presented, which show the effectiveness of the proposed approach.**Description:**C1**Description:**2003001350

**Authors:**Bagirov, Adil**Date:**2005**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 20, no. 2-3 (2005), p. 271-290**Full Text:****Reviewed:****Description:**We consider the problem of discriminating two finite point sets in the n-dimensional space by a finite number of hyperplanes generating a piecewise linear function. If the intersection of these sets is empty, then they can be strictly separated by a max-min of linear functions. An error function is introduced. This function is nonconvex piecewise linear. We discuss an algorithm for its minimization. The results of numerical experiments using some real-world datasets are presented, which show the effectiveness of the proposed approach.**Description:**C1**Description:**2003001350

Discrete gradient method : Derivative-free method for nonsmooth optimization

- Bagirov, Adil, Karasozen, Bulent, Sezer, Monsalve

**Authors:**Bagirov, Adil , Karasozen, Bulent , Sezer, Monsalve**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 137, no. 2 (2008), p. 317-334**Relation:**http://purl.org/au-research/grants/arc/DP0666061**Full Text:****Reviewed:****Description:**A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with the nonsmooth optimization solver DNLP from CONOPT-GAMS and the derivative-free optimization solver CONDOR are presented. © 2007 Springer Science+Business Media, LLC.**Description:**C1

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.

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.

A quasisecant method for minimizing nonsmooth functions

- Bagirov, Adil, Ganjehlou, Asef Nazari

**Authors:**Bagirov, Adil , Ganjehlou, Asef Nazari**Date:**2010**Type:**Text , Journal article**Relation:**Optimization Methods and Software Vol. 25, no. 1 (2010), p. 3-18**Relation:**http://purl.org/au-research/grants/arc/DP0666061**Full Text:**false**Reviewed:****Description:**We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.

Codifferential method for minimizing nonsmooth DC functions

**Authors:**Bagirov, Adil , Ugon, Julien**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 50, no. 1 (2011), p. 3-22**Relation:**http://purl.org/au-research/grants/arc/DP0666061**Full Text:**false**Reviewed:****Description:**In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (DC functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that DC decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given. © 2010 Springer Science+Business Media, LLC.

**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.

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.

Hyperbolic smoothing function method for minimax problems

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

**Authors:**Bagirov, Adil , Al Nuaimat, Alia , Sultanova, Nargiz**Date:**2013**Type:**Text , Journal article**Relation:**Optimization Vol. 62, no. 6 (2013), p. 759-782**Full Text:**false**Reviewed:****Description:**In this article, an approach for solving finite minimax problems is proposed. This approach is based on the use of hyperbolic smoothing functions. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in general algebraic modelling system. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem. © 2013 Copyright Taylor and Francis Group, LLC.**Description:**2003011099

An incremental clustering algorithm based on hyperbolic smoothing

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

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

Nonsmooth optimization 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.

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.

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.

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.

Subgradient Method for Nonconvex Nonsmooth Optimization

- Bagirov, Adil, Jin, L., Karmitsa, Napsu, Al Nuaimat, A., Sultanova, Nargiz

**Authors:**Bagirov, Adil , Jin, L. , Karmitsa, Napsu , Al Nuaimat, A. , Sultanova, Nargiz**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol.157, no.2 (2012), p.416–435**Full Text:**false**Reviewed:****Description:**In this paper, we introduce a new method for solving nonconvex nonsmooth optimization problems. It uses quasisecants, which are subgradients computed in some neighborhood of a point. The proposed method contains simple procedures for finding descent directions and for solving line search subproblems. The convergence of the method is studied and preliminary results of numerical experiments are presented. The comparison of the proposed method with the subgradient and the proximal bundle methods is demonstrated using results of numerical experiments. © 2012 Springer Science+Business Media, LLC.

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