This paper deals with a new and efficient collective optimization approach, based on DC (Difference of Convex functions) programming and DCA (DC Algorithm), powerful tools of nonconvex programming. Exploiting the efficiency and the flexibility of DCA we develop the so-called collaborative DCA in which divers DCA based algorithms are cooperated in an effective way. Two versions of collaborative DCA are proposed and their applications on clustering, a fundamental problem in unsupervised learning, are studied. Numerical experiments are performed on several datasets. The comparative results with three DCA component algorithms show that the collaborative DCA outperforms them on quality and it realizes a good trade-off between the quality of solutions and the running time.
S.Arora and R.Kannan, LearningMixtures of Arbitrary Gaussians, in: Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing, New York, NY, USA, (2001), pp. 247–257.
2.
A.M.Bagirov, Modified global k-means algorithm for minimumsum-of-squares clustering problems, Pattern Recognition41(10) (2008), 3192–3199.
3.
M.J.Brusco, A repetitive branch-and-bound procedure for minimum within-cluster sum of squares partitioning, Psychometrika71(2) (2006), 347–363.
4.
I.Dhillon, J.Kogan and C.Nicholas, Feature Selection and Document Clustering, in: Survey of Text Mining, M.W.Berry, ed., SpringerNew York, 2004, pp. 73–100.
5.
M.Filippone, F.Camastra, F.Masulli and S.Rovetta, A survey of kernel and spectral methods for clustering, Pattern Recognition41(1) (2008), 176–190.
6.
A.K.Jain, Data clustering: 50 years beyond K-means, Pattern Recognition Letters31 (2010), 651–666.
7.
A.K.Jain, M.N.Murty and P.J.Flynn, Data Clustering: A Review, ACM Comput Surv31(3) (1999), 264–323.
8.
J.J.Júdice, M.Raydan, S.S.Rosa and S.A.Santos, On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm, Numerical Algorithms47(4) (2008), 391–407.
9.
H.M.Le, H.A.Le Thi, T.Pham Dinh and V.N.Huynh, Block Clustering Based on Difference of Convex Functions (DC) Programming and DC Algorithms, Neural Comput25(10) (2013), 2776–2807.
10.
H.A.Le Thi, M.T.Belghiti and T.Pham Dinh, A new efficient algorithm based on DC programming and DCA for clustering, Journal of Global Optimization37(4) (2007), 593–608.
11.
H.A.Le Thi, H.M.Le and T.Pham Dinh, Fuzzy clustering based on nonconvex optimisation approaches using difference of convex (DC) functions algorithms, Advances in Data Analysis and Classification1(2) (2007), 85–104.
12.
H.A.Le Thi, H.M.Le and T.Pham Dinh, New and efficient DCA based algorithms for minimum sum-of-squares clustering, Pattern Recognition47(1) (2014), 388–401.
13.
H.A.Le Thi and T.PhamDinh, The DC (Difference of Convex functions) Programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann Oper Res133(1–4) (2005), 23–48.
14.
H.A.Le Thi and T.Pham Dinh, DC programming and DCA: Thirty years of developments, Mathematical programming, Special Issue: DC Programming - Theory, Algorithms and Applications169(1) (2018), 5–68.
15.
T.Pham Dinh and M.Castellani, The Bees Algorithm: Modelling foraging behaviour to solve continuous optimization problems, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science223(12) (2009), 2919–2938.
16.
T.Pham Dinh and H.A.Le Thi, Convex analysis approach to d.c. programming: Theory, Algorithms and Applications, Acta Mathematica Vietnamica22(1) (1997), 289–355, dedicated to Professor Hoang Tuy on the occasion of his 70th birthday.
17.
T.Pham Dinh and H.A.Le Thi, A D.C. Optimization Algorithm for Solving the Trust-Region Subproblem, SIAMJournal on Optimization8(2) (1998), 476–505.
18.
T.Pham Dinh and H.A.Le Thi, Recent Advances in DC Programming and DCA, in: Transactions on Computational Intelligence XIII, N.-T.Nguyen and H.A.Le Thi, eds, LectureNotes in Computer Science, Vol. 8342, Springer Berlin Heidelberg, 2014, pp. 1–37.
19.
S.Sengupta, S.Basak and R.A.Peters, Particle Swarm Optimization: A Survey of Historical and Recent Developments with Hybridization Perspectives, Machine Learning and Knowledge Extraction1(1) (2018), 157–191.
20.
B.K.Sriperumbudur, D.A.Torres and G.R.G.Lanckriet, Sparse Eigen Methods by D.C. Programming, in: Proceedings of the 24th International Conference on Machine Learning, ICML ’07, ACM, New York, NY, USA, 2007, pp. 831–838.
21.
H.D.Vinod, Integer Programming and the Theory of Grouping, Journal of the American Statistical Association64(326) (1969), 506–519.
22.
M.Yagouni and H.A.Le Thi, A Collaborative Metaheuristic Optimization Scheme: Methodological Issues, in: Advanced Computational Methods for Knowledge Engineering, T.van Do, H.A.Le Thi and N.T.Nguyen, eds, Springer International Publishing, 2014, pp. 3–14.
