Subgradient Method for Nonconvex Nonsmooth Optimization
- 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
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- 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.
A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes
- 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
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- 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.
New diagonal bundle method for clustering problems in large data sets
- 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
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- 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.
Clustering in large data sets with the limited memory bundle method
- 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
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- 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
- 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
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- 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
- Authors: Bagirov, Adil , Taheri, Sona , Joki, Kaisa , Karmitsa, Napsu , Mäkelä, Marko
- Date: 2021
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 15, no. 1 (2021), p. 83-96
- Relation: http://purl.org/au-research/grants/arc/DP190100580
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- Description: The aggregate subgradient method is developed for solving unconstrained nonsmooth difference of convex (DC) optimization problems. The proposed method shares some similarities with both the subgradient and the bundle methods. Aggregate subgradients are defined as a convex combination of subgradients computed at null steps between two serious steps. At each iteration search directions are found using only two subgradients: the aggregate subgradient and a subgradient computed at the current null step. It is proved that the proposed method converges to a critical point of the DC optimization problem and also that the number of null steps between two serious steps is finite. The new method is tested using some academic test problems and compared with several other nonsmooth DC optimization solvers. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.