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14Bagirov, Adil
5Karmitsa, Napsu
4Taheri, Sona
4Ugon, Julien
3Gao, David
2Burachik, Regina
2Ganjehlou, Asef Nazari
2Joki, Kaisa
2Kaya, Yalcin
2Makela, Marko
2Mammadov, Musa
2Mirzayeva, Hijran
1Al Nuaimat, A.
1Asadi, Soodabeh
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1Ferguson, Brent
1Hassani, Sara
1Ivkovic, Sasha
1Jamshidi, Mina
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16Nonsmooth optimization
110102 Applied Mathematics
110103 Numerical and Computational Mathematics
40802 Computation Theory and Mathematics
4DC functions
30906 Electrical and Electronic Engineering
3Bundle method
3Bundle methods
3Subdifferential
20101 Pure Mathematics
20801 Artificial Intelligence and Image Processing
2Canonical duality
2Classification
2Cluster analysis
2Cutting plane model
2Optimization
2Separability
2Triality theory
10105 Mathematical Physics

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New diagonal bundle method for clustering problems in large data sets

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

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

A generalized subgradient method with piecewise linear subproblem

- Bagirov, Adil, Ganjehlou, Asef Nazari, Tor, Hakan, Ugon, Julien

**Authors:**Bagirov, Adil , Ganjehlou, Asef Nazari , Tor, Hakan , Ugon, Julien**Date:**2010**Type:**Text , Journal article**Relation:**Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 17, no. 5 (2010), p. 621-638**Full Text:**false**Reviewed:****Description:**In this paper, a new version of the quasisecant method for nonsmooth nonconvex optimization is developed. Quasisecants are overestimates to the objective function in some neighborhood of a given point. Subgradients are used to obtain quasisecants. We describe classes of nonsmooth functions where quasisecants can be computed explicitly. We show that a descent direction with suffcient decrease must satisfy a set of linear inequalities. In the proposed algorithm this set of linear inequalities is solved by applying the subgradient algorithm to minimize a piecewise linear function. We compare results of numerical experiments between the proposed algorithm and subgradient method. Copyright Â© 2010 Watam Press.

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.

Optimality conditions via weak subdifferentials in reflexive Banach spaces

- Hassani, Sara, Mammadov, Musa, Jamshidi, Mina

**Authors:**Hassani, Sara , Mammadov, Musa , Jamshidi, Mina**Date:**2017**Type:**Text , Journal article**Relation:**Turkish Journal of Mathematics Vol. 41, no. 1 (2017), p. 1-8**Full Text:****Reviewed:****Description:**In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. © Tübitak.

**Authors:**Hassani, Sara , Mammadov, Musa , Jamshidi, Mina**Date:**2017**Type:**Text , Journal article**Relation:**Turkish Journal of Mathematics Vol. 41, no. 1 (2017), p. 1-8**Full Text:****Reviewed:****Description:**In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. © Tübitak.

Canonical duality theory and triality for solving general global optimization problems in complex systems

- Morales-Silva, Daniel, Gao, David

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

An update rule and a convergence result for a penalty function method

- Burachik, Regina, Kaya, Yalcin

**Authors:**Burachik, Regina , Kaya, Yalcin**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Industrial & Management Optimization Vol. 3, no. 2 (2007), p. 381-398**Full Text:****Reviewed:****Description:**We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results.**Description:**C1**Description:**2003004886

**Authors:**Burachik, Regina , Kaya, Yalcin**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Industrial & Management Optimization Vol. 3, no. 2 (2007), p. 381-398**Full Text:****Reviewed:****Description:**We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results.**Description:**C1**Description:**2003004886

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

New algorithms for multi-class cancer diagnosis using tumor gene expression signatures

- Bagirov, Adil, Ferguson, Brent, Ivkovic, Sasha, Saunders, Gary, Yearwood, John

**Authors:**Bagirov, Adil , Ferguson, Brent , Ivkovic, Sasha , Saunders, Gary , Yearwood, John**Date:**2003**Type:**Text , Journal article**Relation:**Bioinformatics Vol. 19, no. 14 (2003), p. 1800-1807**Full Text:****Reviewed:****Description:**Motivation: The increasing use of DNA microarray-based tumor gene expression profiles for cancer diagnosis requires mathematical methods with high accuracy for solving clustering, feature selection and classification problems of gene expression data. Results: New algorithms are developed for solving clustering, feature selection and classification problems of gene expression data. The clustering algorithm is based on optimization techniques and allows the calculation of clusters step-by-step. This approach allows us to find as many clusters as a data set contains with respect to some tolerance. Feature selection is crucial for a gene expression database. Our feature selection algorithm is based on calculating overlaps of different genes. The database used, contains over 16 000 genes and this number is considerably reduced by feature selection. We propose a classification algorithm where each tissue sample is considered as the center of a cluster which is a ball. The results of numerical experiments confirm that the classification algorithm in combination with the feature selection algorithm perform slightly better than the published results for multi-class classifiers based on support vector machines for this data set.**Description:**C1**Description:**2003000439

**Authors:**Bagirov, Adil , Ferguson, Brent , Ivkovic, Sasha , Saunders, Gary , Yearwood, John**Date:**2003**Type:**Text , Journal article**Relation:**Bioinformatics Vol. 19, no. 14 (2003), p. 1800-1807**Full Text:****Reviewed:****Description:**Motivation: The increasing use of DNA microarray-based tumor gene expression profiles for cancer diagnosis requires mathematical methods with high accuracy for solving clustering, feature selection and classification problems of gene expression data. Results: New algorithms are developed for solving clustering, feature selection and classification problems of gene expression data. The clustering algorithm is based on optimization techniques and allows the calculation of clusters step-by-step. This approach allows us to find as many clusters as a data set contains with respect to some tolerance. Feature selection is crucial for a gene expression database. Our feature selection algorithm is based on calculating overlaps of different genes. The database used, contains over 16 000 genes and this number is considerably reduced by feature selection. We propose a classification algorithm where each tissue sample is considered as the center of a cluster which is a ball. The results of numerical experiments confirm that the classification algorithm in combination with the feature selection algorithm perform slightly better than the published results for multi-class classifiers based on support vector machines for this data set.**Description:**C1**Description:**2003000439

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.

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.

Clustering in large data sets with the limited memory bundle method

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

**Authors:**Karmitsa, Napsu , Bagirov, Adil , Taheri, Sona**Date:**2018**Type:**Text , Journal article**Relation:**Pattern Recognition Vol. 83, no. (2018), p. 245-259**Relation:**http://purl.org/au-research/grants/arc/DP140103213**Full Text:**false**Reviewed:****Description:**The aim of this paper is to design an algorithm based on nonsmooth optimization techniques to solve the minimum sum-of-squares clustering problems in very large data sets. First, the clustering problem is formulated as a nonsmooth optimization problem. Then the limited memory bundle method [Haarala et al., 2007] is modified and combined with an incremental approach to design a new clustering algorithm. The algorithm is evaluated using real world data sets with both the large number of attributes and the large number of data points. It is also compared with some other optimization based clustering algorithms. The numerical results demonstrate the efficiency of the proposed algorithm for clustering in very large data sets.

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

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

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

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

Nonsmooth optimization based algorithms in cluster analysis

- Bagirov, Adil, Mohebi, Ehsan

**Authors:**Bagirov, Adil , Mohebi, Ehsan**Date:**2015**Type:**Text , Book chapter**Relation:**Partitional Clustering Algorithms p. 99-146**Full Text:**false**Reviewed:****Description:**Cluster analysis is an important task in data mining. It deals with the problem of organization of a collection of objects into clusters based on a similarity measure. Various distance functions can be used to define the similarity measure. Cluster analysis problems with the similarity measure defined by the squared Euclidean distance, which is also known as the minimum sum-of-squares clustering, has been studied extensively over the last five decades. L1 and L1 norms have attracted less attention. In this chapter, we consider a nonsmooth nonconvex optimization formulation of the cluster analysis problems. This formulation allows one to easily apply similarity measures defined using different distance functions. Moreover, an efficient incremental algorithm can be designed based on this formulation to solve the clustering problems. We develop incremental algorithms for solving clustering problems where the similarity measure is defined using the L1; L2 and L1 norms. We also consider different algorithms for solving nonsmooth nonconvex optimization problems in cluster analysis. The proposed algorithms are tested using several real world data sets and compared with other similar algorithms.**Description:**Cluster analysis is an important task in data mining. It deals with the problem of organization of a collection of objects into clusters based on a similarity measure. Various distance functions can be used to define the similarity measure. Cluster analysis problems with the similarity measure defined by the squared Euclidean distance, which is also known as the minimum sum-of-squares clustering, has been studied extensively over the last five decades. However, problems with the L

An inexact modified subgradient algorithm for nonconvex optimization

- Burachik, Regina, Kaya, Yalcin, Mammadov, Musa

**Authors:**Burachik, Regina , Kaya, Yalcin , Mammadov, Musa**Date:**2008**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. , no. (2008), p. 1-24**Full Text:****Reviewed:****Description:**We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang-bang control problem, under several different inexactness schemes. © 2008 Springer Science+Business Media, LLC.**Description:**C1

**Authors:**Burachik, Regina , Kaya, Yalcin , Mammadov, Musa**Date:**2008**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. , no. (2008), p. 1-24**Full Text:****Reviewed:****Description:**We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang-bang control problem, under several different inexactness schemes. © 2008 Springer Science+Business Media, LLC.**Description:**C1

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.

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.

On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

- Yuan, Y. B., Fang, Shucherng, Gao, David

**Authors:**Yuan, Y. B. , Fang, Shucherng , Gao, David**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Global Optimization Vol. 52, no. 2 (2012), p. 195-209**Full Text:**false**Reviewed:****Description:**This paper studies the canonical duality theory for solving a class of quadri- nomial minimization problems subject to one general quadratic constraint. It is shown that the nonconvex primal problem in Rn can be converted into a concave maximization dual problem over a convex set in R2 , such that the problem can be solved more efficiently. The existence and uniqueness theorems of global minimizers are provided using the triality theory. Examples are given to illustrate the results obtained. © 2011 Springer Science+Business Media, LLC.

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**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.

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