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Mathematical Programming

, Volume 169, Issue 1, pp 5–68 | Cite as

DC programming and DCA: thirty years of developments

  • Hoai An Le Thi
  • Tao Pham Dinh
Full Length Paper Series B

Abstract

The year 2015 marks the 30th birthday of DC (Difference of Convex functions) programming and DCA (DC Algorithms) which constitute the backbone of nonconvex programming and global optimization. In this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools. The survey is comprised of three parts. In the first part we present a brief history of the field, while in the second we summarize the state-of-the-art results and recent advances. We focus on main theoretical results and DCA solvers for important classes of difficult nonconvex optimization problems, and then give an overview of real-world applications whose solution methods are based on DCA. The third part is devoted to new trends and important open issues, as well as suggestions for future developments.

Keywords

DC programming DCA Theory Algorithms Applications 

Mathematics Subject Classification

90C26 90C90 

Notes

Acknowledgements

The authors are grateful to Dr. Vo Xuan Thanh for sending us some references on DCA solvers for real-world applications, and the two anonymous reviewers as well as Professor Jong-Shi Pang for their constructive comments that greatly improved the manuscript, in particular one of reviewers for providing us some references on related DCA methods in Sect. 3.3.

References

  1. 1.
    Ahn, M., Pang, J.S., Xin, J.: Difference-of-convex learning: directional stationarity, optimality, and sparsity. SIAM J. Optim. 27(3), 1637–1665 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Akoa, F.B.: Combining DC algorithms (DCAs) and decomposition techniques for the training of nonpositive-semidefinite kernels. IEEE Trans. Neural Netw. 19(11), 1854–1872 (2008)CrossRefGoogle Scholar
  3. 3.
    Alexandroff, A.: On functions representable as a difference of convex functions. Doklady Akad. Nauk SSSR (N.S.) 72, 613–616 . [English translation: Siberian Elektron. Mathetical. Izv. 9 (2012) 360–376.] (1950)Google Scholar
  4. 4.
    Alvarado, A., Scutari, G., Pang, J.S.: A new decomposition method for multiuser dc-programming and its applications. IEEE Trans. Signal Process. 62(11), 2984–2998 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Argyriou, A., Hauser, R., Micchelli, C.A., Pontil, M.: A DC-programming algorithm for kernel selection. In: ICML 2006, pp. 41–48. ACM (2006)Google Scholar
  6. 6.
    Arthanari, T.S., Le Thi, H.A.: New formulations of the multiple sequence alignment problem. Optim. Lett. 5(1), 27–40 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Astorino, A., Fuduli, A.: Semisupervised spherical separation. Appl. Math. Model. 39(20), 6351–6358 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Astorino, A., Fuduli, A., Gaudioso, M.: DC models for spherical separation. J. Global Optim. 48(4), 657–669 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Astorino, A., Fuduli, A., Gaudioso, M.: Margin maximization in spherical separation. Comput. Optim. Appl. 53(2), 301–322 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116(1), 5–16 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137(1), 91–129 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bačák, M., Borwein, J.M.: On difference convexity of locally lipschitz functions. Optimization 60(8–9), 961–978 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  15. 15.
    Bertsekas, D.P.: Incremental proximal methods for large scale convex optimization. Math. Program. 129(2), 163–195 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bien, J., Tibshirani, R.J.: Sparse estimation of a covariance matrix. Biometrika 98(4), 807–820 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bottou, L.: On-line learning in neural networks. Chap. In: On-line Learning and Stochastic Approximations, pp. 9–42. Cambridge University Press, New York, NY, USA (1998)Google Scholar
  18. 18.
    Bouallagui, S.: Techniques d’optimisation déterministe et stochastique pour la résolution de problèmes difficiles en cryptologie. Ph.D. thesis, INSA de Rouen (2010)Google Scholar
  19. 19.
    Bouallagui, S., Le Thi, H.A.: Pham Dinh, T.: Design of highly nonlinear balanced boolean functions using an hybridation of DCA and simulated annealing algorithm. In: Modelling, Computation and Optimization in Information Systems and Management Sciences, Communications in Computer and Information Science, vol. 14, pp. 579–588. Springer, Berlin, Heidelberg (2008)Google Scholar
  20. 20.
    Bradley, P.S., Mangasarian, O.L.: Feature selection via concave minimization and support vector machines. ICML 1998, 82–90 (1998)Google Scholar
  21. 21.
    Candes, E.J., Wakin, M., Boyd, S.: Enhancing sparsity by reweighted-\(l_{1}\) minimization. J. Fourier Anal. Appl. 14, 877–905 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Chambolle, A., Vore, R.A.D., Lee, N.Y., Lucier, B.J.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: IEEE International Conference on Acoustics, Speech and Signal Processing, 2008, pp. 3869–3872 (2008)Google Scholar
  24. 24.
    Che, E., Tuan, H.D., Nguyen, H.H.: Joint optimization of cooperative beamforming and relay assignment in multi-user wireless relay networks. IEEE Trans. Wirel. Commun. 13(10), 5481–5495 (2014)CrossRefGoogle Scholar
  25. 25.
    Chen, G., Zeng, D., Kosorok, M.R.: Personalized dose finding using outcome weighted learning. J. Am. Stat. Assoc. 111(516), 1509–1521 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Cheng, Y., Pesavento, M.: Joint optimization of source power allocation and distributed relay beamforming in multiuser peer-to-peer relay networks. IEEE Trans. Signal Process. 60(6), 2962–2973 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Cheung, P.M., Kwok, J.T.: A regularization framework for multiple-instance learning. In: ICML 2006, pp. 193–200. ACM, New York, NY, USA (2006)Google Scholar
  28. 28.
    Collobert, R., Sinz, F., Weston, J., Bottou, L.: Large scale transductive SVMs. J. Mach. Learn. Res. 7, 1687–1712 (2006)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Collobert, R., Sinz, F., Weston, J., Bottou, L.: Trading convexity for scalability. In: ICML 2006, pp. 201–208 (2006)Google Scholar
  30. 30.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Conn, A., Gould, N., Toint, P.: Trust Region Methods. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  32. 32.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B Methodol. 39(1), 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  34. 34.
    El Azami, M., Lartizien, C., Canu, S.: Robust outlier detection with L0-SVDD. In: European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning, ESANN 2014, pp. 389–394 (2014)Google Scholar
  35. 35.
    Ellis, S.E., Nayakkankuppam, M.V.: Phylogenetic analysis via DC programming . (Preprint) (2003)Google Scholar
  36. 36.
    Esser, E., Lou, Y., Xin, J.: A method for finding structured sparse solutions to nonnegative least squares problems with applications. SIAM J. Imaging Sci. 6(4), 2010–2046 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Fastrich, B., Paterlini, S., Winker, P.: Constructing optimal sparse portfolios using regularization methods. CMS 12(3), 417–434 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Fawzi, A., Davies, M., Frossard, P.: Dictionary learning for fast classification based on soft-thresholding. Int. J. Comput. Vis. 114(2), 306–321 (2015)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Feng, D., Yu, G., Yuan-Wu, Y., Li, G.Y., Feng, G., Li, S.: Mode switching for energy-efficient device-to-device communications in cellular networks. IEEE Trans. Wirel. Commun. 14(12), 6993–7003 (2015)CrossRefGoogle Scholar
  41. 41.
    Floudas, C.A., Pardalos, P.M., Adjiman, C., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of test problems in local and global optimization. In: Nonconvex Optimization and Its Applications, vol. 33. Springer, USA (1999)Google Scholar
  42. 42.
    Gasso, G., Pappaioannou, A., Spivak, M., Bottou, L.: Batch and online learning algorithms for nonconvex Neyman–Pearson classification. ACM Trans. Intell. Syst. Technol. 2(3), 28:1–28:19 (2011)CrossRefGoogle Scholar
  43. 43.
    Gasso, G., Rakotomamonjy, A., Canu, S.: Recovering sparse signals with a certain family of nonconvex penalties and DC programming. IEEE Trans. Signal Process. 57(12), 4686–4698 (2009)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Geng, J., Wang, L., Wang, Y.: A non-convex algorithm framework based on DC programming and DCA for matrix completion. Numer. Algorithms 68(4), 903–921 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Gholami, M.R., Gezici, S., Strom, E.G.: A concave–convex procedure for TDOA based positioning. IEEE Commun. Lett. 17(4), 765–768 (2013)CrossRefGoogle Scholar
  46. 46.
    Göernitz, N., Braun, M., Kloft, M.: Hidden Markov anomaly detection. In: Proceedings of the 32nd International Conference on Machine Learning, vol. 37, pp. 1833–1842. JMLR: W&CP (2015)Google Scholar
  47. 47.
    Gong, P., Zhang, C., Lu, Z., Huang, J.Z., Ye, J.: A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems. In: Proceedings of the 30th International Conference on International Conference on Machine Learning, ICML’13, vol. 28, pp. II-37–II-45 (2013)Google Scholar
  48. 48.
    Gorodnitsky, I.F., Rao, B.D.: Sparse signal reconstructions from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Signal Process. 45(3), 600–616 (1997)CrossRefGoogle Scholar
  49. 49.
    Guan, G., Gray, A.: Sparse high-dimensional fractional-norm support vector machine via DC programming. Comput. Stat. Data Anal. 67, 136–148 (2013)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Gülpinar, N., Le Thi, H.A., Moeini, M.: Robust investment strategies with discrete asset choice constraints using DC programming and DCA. Optimization 59(1), 45–62 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation for \(\ell _1\)-minimization: methodology and convergence. SIAM J. Optim. 19(3), 1107–1130 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math. 9(3), 707–713 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Heinkenschloss, M.: On the solution of a two ball trust region subproblem. Math. Program. 64(1–3), 249–276 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Hiriart-Urruty, J.B.: From Convex Optimization to Nonconvex Optimization. Part I Necessary and Sufficient Conditions for Global Optimality, pp. 219–239. Springer, Boston (1989)zbMATHGoogle Scholar
  55. 55.
    Ho, V.T.: Advanced machine learning techniques based on DC programming and DCA. Ph.D. thesis, University of Lorraine (2017)Google Scholar
  56. 56.
    Ho, V.T., Le Thi, H.A.: Solving an infinite-horizon discounted Markov decision process by DC programming and DCA. In: Nguyen, T.B., van Do, T., Le Thi, H.A., Nguyen, N.T. (eds.) Advanced Computational Methods for Knowledge Engineering: ICCSAMA 2016, Proceedings, Part I, pp. 43–55. Springer, Berlin (2016)CrossRefGoogle Scholar
  57. 57.
    Ho, V.T., Le Thi, H.A., Bui, D.C.: Online DC optimization for online binary linear classification. In: Nguyen, T.N., Trawiński, B., Fujita, H., Hong, T.P. (eds.) Intelligent Information and Database Systems: ACIIDS 2016, Proceedings, Part II, pp. 661–670. Springer, Berlin (2016)Google Scholar
  58. 58.
    Hong, M., Razaviyayn, M., Luo, Z.Q., Pang, J.S.: A unified algorithmic framework for block-structured optimization involving big data: with applications in machine learning and signal processing. IEEE Signal Process. Mag. 33(1), 57–77 (2016)CrossRefGoogle Scholar
  59. 59.
    Huang, X., Shi, L., Suykens, J.: Ramp loss linear programming support vector machine. J. Mach. Learn. Res. 15(1), 2185–2211 (2014)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Hunter, D.R., Lange, K.: Rejoinder to discussion of optimization transfer using surrogate objective functions. Comput. Graph. Stat. 9, 52–59 (2000)Google Scholar
  61. 61.
  62. 62.
    Jara-Moroni, F., Pang, J.S., Waechter, A.: A study of the difference-of-convex approach for solving linear programs with complementarity constraints. Math. Program. Ser. B (2018, to appear)Google Scholar
  63. 63.
    Jeong, S., Simeone, O., Haimovich, A., Kang, J.: Optimal fronthaul quantization for cloud radio positioning. IEEE Trans. Veh. Technol. 65(4), 2763–2768 (2016)CrossRefGoogle Scholar
  64. 64.
    Júdice, J.J., Sherali, H.D., Ribeiro, I.M.: The eigenvalue complementarity problem. Comput. Optim. Appl. 37(2), 139–156 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Júdice, J.J., Sherali, H.D., Ribeiro, I.M., Rosa, S.S.: On the asymmetric eigenvalue complementarity problem. Optim. Methods Softw. 24(4–5), 549–568 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Global Optim. 13(4), 389–406 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Khalaf, W., Astorino, A., D’Alessandro, P., Gaudioso, M.: A DC optimization-based clustering technique for edge detection. Optim. Lett. 11(3), 627–640 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Kim, S., Pan, W., Shen, X.: Network-based penalized regression with application to genomic data. Biometrics 69(3), 582–93 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Krause, N., Singer, Y.: Leveraging the margin more carefully. In: Proceedings of the twenty-first international conference on Machine learning ICML 2004, p. 63 (2004)Google Scholar
  70. 70.
    Krummenacher, G., Ong, C.S., Buhmann, J.: Ellipsoidal multiple instance learning. In: Dasgupta, S., Mcallester, D. (eds.) ICML 2013, JMLR: W&CP, vol. 28, pp. 73–81 (2013)Google Scholar
  71. 71.
    Kuang, Q., Speidel, J., Droste, H.: Joint base-station association, channel assignment, beamforming and power control in heterogeneous networks. In: IEEE 75th Vehicular Technology Conference (VTC Spring), pp. 1–5 (2012)Google Scholar
  72. 72.
    Kwon, S., Ahn, J., Jang, W., Lee, S., Kim, Y.: A doubly sparse approach for group variable selection. Ann. Inst. Stat. Math. 69(5), 997–1025 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Laporte, L., Flamary, R., Canu, S., Déjean, S., Mothe, J.: Nonconvex regularizations for feature selection in ranking with sparse SVM. IEEE Trans. Neural Netw. Learn. 25(6), 1118–1130 (2014)CrossRefGoogle Scholar
  74. 74.
    Le, A.V., Le Thi, H.A., Nguyen, M.C., Zidna, A.: Network intrusion detection based on multi-class support vector machine. In: Nguyen, N.T., Hoang, K., Jedrzejowicz, P. (eds.) Computational Collective Intelligence. Technologies and Applications: ICCCI 2012, Proceedings, Part I, pp. 536–543. Springer, Berlin (2012)CrossRefGoogle Scholar
  75. 75.
    Le, H.M.: Modélisation et optimisation non convexe basées sur la programmation DC et DCA pour la résolution de certaines classes des problémes en fouille de données et cryptologie. Ph.D. thesis, Université Paul Verlaine-Metz (2007)Google Scholar
  76. 76.
    Le, H.M., Le Thi, H.A., Nguyen, M.C.: Sparse semi-supervised support vector machines by DC programming and DCA. Neurocomputing 153, 62–76 (2015)CrossRefGoogle Scholar
  77. 77.
    Le, H.M., Le Thi, H.A., Pham Dinh, T., Bouvry, P.: A combined DCA: GA for constructing highly nonlinear balanced boolean functions in cryptography. J. Global Optim. 47(4), 597–613 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Le, H.M., Le Thi, H.A., Pham Dinh, T., Huynh, V.N.: Block clustering based on Difference of Convex functions (DC) programming and DC algorithms. Neural Comput. 25(10), 2776–2807 (2013)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Le, H.M., Nguyen, T.B.T., Ta, M.T., Le Thi, H.A.: Image segmentation via feature weighted fuzzy clustering by a DCA based algorithm. In: Advanced Computational Methods for Knowledge Engineering, Studies in Computational Intelligence, vol. 479, pp. 53–63. Springer (2013)Google Scholar
  80. 80.
    Le, H.M., Ta, M.T.: DC programming and DCA for solving minimum sum-of-squares clustering using weighted dissimilarity measures. In: Transactions on Computational Intelligence XIII, LNCS, vol. 8342, pp. 113–131. Springer, Berlin, Heidelberg (2014)Google Scholar
  81. 81.
    Le, H.M., Yassine, A., Moussi, R.: DCA for solving the scheduling of lifting vehicle in an automated port container terminal. Comput. Manag. Sci. 9(2), 273–286 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Le Thi, H.A.: Analyse numérique des algorithmes de l’optimisation DC. Approches locale et globale. Codes et simulations numériques en grande dimension. Applications. Ph.D. thesis, Université de Rouen (1994)Google Scholar
  83. 83.
    Le Thi, H.A.: Contribution à l’optimisation non convexe et l’optimisation globale: : Théorie. Algorithmes et Applications. Habilitation à Diriger des Recherches, National Institute for Applied Sciences, Rouen (1997)Google Scholar
  84. 84.
    Le Thi, H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Math. Program. 87(3), 401–426 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Le Thi, H.A.: Solving large scale molecular distance geometry problems by a smoothing technique via the Gaussian transform and D.C. programming. J. Global Optim. 27(4), 375–397 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Le Thi, H.A.: DCA collaborative for clustering. University of Lorraine, Tech. rep. (2013)Google Scholar
  87. 87.
    Le Thi, H.A.: Phylogenetic tree reconstruction by a DCA based algorithm. Tech. rep., LITA, University of Lorraine (2013)Google Scholar
  88. 88.
    Le Thi, H.A.: DC programming and DCA for challenging problems in bioinformatics and computational biology. In: Adamatzky, A. (ed.) Automata, Universality, Computation, Emergence, Complexity and Computation, vol. 12, pp. 383–414. Springer, Berlin (2015)Google Scholar
  89. 89.
    Le Thi, H.A., Belghiti, M.T., Pham Dinh, T.: A new efficient algorithm based on DC programming and DCA for clustering. J. Global Optim. 37(4), 593–608 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Le Thi, H.A., Ho, V.T.: Online learning based on Online DCA (2016, Submitted)Google Scholar
  91. 91.
    Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: DC programming and DCA for general DC programs. In: van Do, T., Le Thi, H.A., Nguyen, N.T. (eds.) Advanced Computational Methods for Knowledge Engineering, pp. 15–35. Springer, Berlin (2014)zbMATHCrossRefGoogle Scholar
  92. 92.
    Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: Error bounds via exact penalization with applications to concave and quadratic systems. J. Optim. Theory Appl. 171(1), 228–250 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: Convergence analysis of DCA with subanalytic data. J. Optim. Theory Appl. (2018)Google Scholar
  94. 94.
    Le Thi, H.A., Huynh, V.N., Pham Dinh, T., Vaz, A.I.F., Vicente, L.N.: Globally convergent DC trust-region methods. J. Global Optim. 59(2), 209–225 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Le Thi, H.A., Le, H.M., Nguyen, T.P., Pham Dinh, T.: Noisy image segmentation by a robust clustering algorithm based on DC programming and DCA. In: Proceedings of the 8th Industrial Conference on Advances in Data Mining, ICDM’08, pp. 72–86. Springer (2008)Google Scholar
  96. 96.
    Le Thi, H.A., Le, H.M., Nguyen, V.V., Pham Dinh, T.: A DC programming approach for feature selection in support vector machines learning. Adv. Data Anal. Classif. 2(3), 259–278 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Le Thi, H.A., Le, H.M., Pham Dinh, T.: Fuzzy clustering based on nonconvex optimisation approaches using difference of convex (DC) functions algorithms. Adv. Data Anal. Classif. 1(2), 85–104 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Le Thi, H.A., Le, H.M., Pham Dinh, T.: New and efficient DCA based algorithms for minimum sum-of-squares clustering. Pattern Recognit. 47(1), 388–401 (2014)zbMATHCrossRefGoogle Scholar
  99. 99.
    Le Thi, H.A., Le, H.M., Pham Dinh, T.: Feature selection in machine learning: an exact penalty approach using a difference of convex function algorithm. Mach. Learn. 101(1–3), 163–186 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Le Thi, H.A., Le, H.M., Pham Dinh, T., Bouvry, P.: Solving the perceptron problem by deterministic optimization approach based on DC programming and DCA. In: INDIN 2009, Cardiff. IEEE (2009)Google Scholar
  101. 101.
    Le Thi, H.A., Le, H.M., Pham Dinh, T., Huynh, V.N.: Binary classification via spherical separator by DC programming and DCA. J. Global Optim. 56(4), 1393–1407 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Le Thi, H.A., Le, H.M., Phan, D.N., Tran, B.: Stochastic DCA for the large-sum of non-convex functions problem and its application to group variable selection in classification. In: Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6–11 August 2017, pp. 3394–3403 (2017)Google Scholar
  103. 103.
    Le Thi, H.A., Le, : M.T., Nguyen, T.B.T.: A novel approach to automated cell counting based on a difference of convex functions algorithm (DCA). In: Computational Collective Intelligence. Technologies and Applications, LNCS, vol. 8083, pp. 336–345. Springer, Berlin, Heidelberg (2013)Google Scholar
  104. 104.
    Le Thi, H.A., Moeini, M.: Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm. J. Optim. Theory Appl. 161(1), 199–224 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    Le Thi, H.A., Moeini, M., Pham Dinh, T.: DC programming approach for portfolio optimization under step increasing transaction costs. Optimization 58(3), 267–289 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Le Thi, H.A., Moeini, M., Pham Dinh, T.: Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA. Comput. Manag. Sci. 6(4), 459–475 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    Le Thi, H.A., Moeini, M., Pham Dinh, T., Joaquim, J.: A DC programming approach for solving the symmetric eigenvalue complementarity problem. Comput. Optim. Appl. 51(3), 1097–1117 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    Le Thi, H.A., Ndiaye, B.M., Pham Dinh, T.: Solving a multimodal transport problem by DCA. In: IEEE International Conference on Research, Innovation and Vision for the Future, pp. 49–56 (2008)Google Scholar
  109. 109.
    Le Thi, H.A., Nguyen, D.M., Pham Dinh, T.: A DC programming approach for planning a multisensor multizone search for a target. Comput. Oper. Res. 41, 231–239 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    Le Thi, H.A., Nguyen, M.C.: Self-organizing maps by difference of convex functions optimization. Data Min. Knowl. Disc. 28(5–6), 1336–1365 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Le Thi, H.A., Nguyen, M.C.: DCA based algorithms for feature selection in multi-class support vector machine. Ann. Oper. Res. 249(1), 273–300 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Le Thi, H.A., Nguyen, M.C., Pham Dinh, T.: A DC programming approach for finding communities in networks. Neural Comput. 26(12), 2827–2854 (2014)MathSciNetCrossRefGoogle Scholar
  113. 113.
    Le Thi, H.A., Nguyen, Q.T.: A robust approach for nonlinear UAV task assignment problem under uncertainty. Transactions on Computational Collective Intelligence II. LNCS, vol. 6450, pp. 147–159. Springer, Berlin, Heidelberg (2010)Google Scholar
  114. 114.
    Le Thi, H.A., Nguyen, Q.T., Nguyen, H.T., Pham Dinh, T.: Solving the earliness tardiness scheduling problem by DC programming and DCA. Math. Balk. 23(3–4), 271–288 (2009)MathSciNetzbMATHGoogle Scholar
  115. 115.
    Le Thi, H.A., Nguyen, Q.T., Nguyen, H.T., Pham Dinh, T.: A time-indexed formulation of earliness tardiness scheduling via DC programming and DCA. In: International Multiconference on Computer Science and Information Technology IMCSIT’09, pp. 2009 (779–784)Google Scholar
  116. 116.
    Le Thi, H.A., Nguyen, Q.T., Phan, K.T., Pham Dinh, T.: Energy minimization-based cross-layer design in wireless networks. In: Proceedings of the 2008 High Performance Computing & Simulation Conference (HPCS 2008), pp. 283–289 (2008)Google Scholar
  117. 117.
    Le Thi, H.A., Nguyen, Q.T., Phan, K.T., Pham Dinh, T.: DC programming and DCA based cross-layer optimization in multi-hop TDMA networks. Intelligent Information and Database Systems. LNCS, vol. 7803, pp. 398–408. Springer, Berlin, Heidelberg (2013)Google Scholar
  118. 118.
    Le Thi, H.A., Nguyen, T.B.T., Le, : H.M.: Sparse signal recovery by difference of convex functions algorithms. In; Intelligent Information and Database Systems. LNCS, vol. 7803, pp. 387–397. Springer, Berlin, Heidelberg (2013)Google Scholar
  119. 119.
    Le Thi, H.A., Nguyen, T.P., Pham Dinh, T.: A continuous DC programming approach to the strategic supply chain design problem from qualified partner set. Eur. J. Oper. Res. 183(3), 1001–1012 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Le Thi, H.A., Nguyen, V.V., Ouchani, S.: Gene selection for cancer classification using DCA. J. Front. Comput. Sci. Technol. 3(6), 612–620 (2009)Google Scholar
  121. 121.
    Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by D.C. algorithms. J. Global Optim. 11(3), 253–285 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  122. 122.
    Le Thi, H.A., Pham Dinh, T.: A branch-and-bound method via D.C. optimization algorithm and ellipsoidal technique for box constrained nonconvex quadratic programming problems. J. Global Optim. 13(2), 171–206 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Le Thi, H.A., Pham Dinh, T.: D.C. programming approach for large-scale molecular optimization via the general distance geometry problem. In: Floudas, C.A., Pardalos, P.M. (eds.) Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, Nonconvex Optimization and Its Applications, vol. 40, pp. 301–339. Springer, New York (2000)CrossRefGoogle Scholar
  124. 124.
    Le Thi, H.A., Pham Dinh, T.: A continuous approach for globally solving linearly constrained quadratic zero-one programming problems. Optimization 50(1–2), 93–120 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    Le Thi, H.A., Pham Dinh, T.: D.C. optimization approaches via Markov models for restoration of signal (1-D) and (2-D). In: Hadjisavvas, N., Pardalos, P. (eds.) Advances in Convex Analysis and Global Optimization, pp. 303–317. Kluwer, Dordrecht (2001)zbMATHGoogle Scholar
  126. 126.
    Le Thi, H.A., Pham Dinh, T.: D.C. programming approach to the multidimensional scaling problem. In: Migdalas, A., Pardalos, P.M., Värbrand, P. (eds.) From Local to Global Optimization, pp. 231–276. Springer, New York (2001)zbMATHGoogle Scholar
  127. 127.
    Le Thi, H.A., Pham Dinh, T.: D.C. programming approach for multicommodity network optimization problems with step increasing cost functions. J. Global Optim. 22(1), 205–232 (2002)MathSciNetzbMATHGoogle Scholar
  128. 128.
    Le Thi, H.A., Pham Dinh, T.: Large scale molecular optimization from distance matrices by a D.C. optimization approach. SIAM J. Optim. 14(1), 77–114 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    Le Thi, H.A., Pham Dinh, T.: A new algorithm for solving large scale molecular distance geometry problems. In: Di Pillo, G., Murli, A. (eds.) High Performance Algorithms and Software for Nonlinear Optimization. Applied Optimization, vol. 82, pp. 285–302. Springer, New York (2003)CrossRefGoogle Scholar
  130. 130.
    Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133(1–4), 23–48 (2005)MathSciNetzbMATHGoogle Scholar
  131. 131.
    Le Thi, H.A., Pham Dinh, T.: A continuous approach for the concave cost supply problem via DC programming and DCA. Discrete Appl. Math. 156(3), 325–338 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  132. 132.
    Le Thi, H.A., Pham Dinh, T.: On solving linear complemetarity problems by DC programming and DCA. Comput. Optim. Appl. 50(3), 507–524 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Le Thi, H.A., Pham Dinh, T.: A two phases DCA based algorithm for solving the Lennard–Jones problem. Tech. rep., LITA, University of Metz (2011)Google Scholar
  134. 134.
    Le Thi, H.A., Pham Dinh, T.: Minimizing the morse potential energy function by a DC programming approach. Tech. rep., LITA, University of Lorraine (2012)Google Scholar
  135. 135.
    Le Thi, H.A., Pham Dinh, T.: DC programming approaches for distance geometry problems. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds.) Distance Geometry: Theory, Methods, and Applications, pp. 225–290. Springer, New York (2013)zbMATHCrossRefGoogle Scholar
  136. 136.
    Le Thi, H.A., Pham Dinh, T.: Network utility maximisation: A DC programming approach for Sigmoidal utility function. In: International Conference on Advanced Technologies for Communications (ATC’13), pp. 50–54 (2013)Google Scholar
  137. 137.
    Le Thi, H.A., Pham Dinh, T.: DC programming in communication systems: challenging problems and methods. Vietnam J. Comput. Sci. 1(1), 15–28 (2014)CrossRefGoogle Scholar
  138. 138.
    Le Thi, H.A., Pham Dinh, T.: Difference of convex functions algorithms (DCA) for image restoration via a Markov random field model. Optim. Eng. 18(4), 873–906 (2017)MathSciNetCrossRefGoogle Scholar
  139. 139.
    Le Thi, H.A., Pham Dinh, T., Belghiti, M.: DCA based algorithms for multiple sequence alignment (MSA). Cent. Eur. J. Oper. Res. 22(3), 501–524 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Le Thi, H.A., Pham Dinh, T., Bouallagui, S.: Cryptanalysis of an identification scheme based on the perceptron problem using a hybridization of deterministic optimization and genetic algorithm. In: Proceedings of the 2009 International Conference on Security & Management, SAM 2009, pp. 117–123. CSREA Press (2009)Google Scholar
  141. 141.
    Le Thi, H.A., Pham Dinh, T., Huynh, V.N.: Exact penalty techniques in DC programming. Tech. rep, National Institute for Applied Sciences, Rouen (2005)Google Scholar
  142. 142.
    Le Thi, H.A., Pham Dinh, T., Huynh, V.N.: Optimization based DC programming and DCA for hierarchical clustering. Eur. J. Oper. Res. 183(3), 1067–1085 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    Le Thi, H.A., Pham Dinh, T., Huynh, V.N.: Exact penalty and error bounds in DC programming. J. Global Optim. 52(3), 509–535 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  144. 144.
    Le Thi, H.A., Pham Dinh, T., Le, H.M., Vo, X.T.: DC approximation approaches for sparse optimization. Eur. J. Oper. Res. 244(1), 26–46 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  145. 145.
    Le Thi, H.A., Pham Dinh, T., Muu, L.D.: Numerical solution for optimization over the efficient set by D.C. optimization algorithm. Oper. Res. Lett. 19(3), 117–128 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  146. 146.
    Le Thi, H.A., Pham Dinh, T., Muu, L.D.: A combined D.C. optimization-ellipsoidal branch-and-bound algorithm for solving nonconvex quadratic programming problems. J. Comb. Optim. 2(1), 9–28 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  147. 147.
    Le Thi, H.A., Pham Dinh, T., Muu, L.D.: Exact penalty in DC programming. Vietnam J. Math. 27(2), 169–179 (1999)MathSciNetzbMATHGoogle Scholar
  148. 148.
    Le Thi, H.A., Pham Dinh, T., Muu, L.D.: Simplicially constrained D.C. optimization over the efficient and weakly efficient sets. J. Optim. Theory Appl. 117(3), 503–521 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  149. 149.
    Le Thi, H.A., Pham Dinh, T., Thiao, M.: Efficient approaches for \(\ell _2-\ell _0\) regularization and applications to feature selection in SVM. Appl. Intell. 45(2), 549–565 (2016)CrossRefGoogle Scholar
  150. 150.
    Le Thi, H.A., Pham Dinh, T., Thoai, N.V.: Combination between global and local methods for solving an optimization problem over the efficient set. Eur. J. Oper. Res. 142(2), 258–270 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  151. 151.
    Le Thi, H.A., Pham Dinh, T., Thoai, N.V., Nguyen Canh, N.: D.C. optimization techniques for solving a class of nonlinear bilevel programs. J. Global Optim. 44(3), 313–337 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  152. 152.
    Le Thi, H.A., Pham Dinh, T., Tran, D.Q.: A DC programming approach for a class of bilevel programming problems and its application in portfolio selection. NACO Numer. Algebra Control Optim. 2(1), 167–185 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  153. 153.
    Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Behavior of DCA sequences for solving the trust-region subproblem. J. Global Optim. 53, 317–329 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  154. 154.
    Le Thi, H.A., Phan, D.N.: DC programming and DCA for sparse optimal scoring problem. Neurocomputing 186, 170–181 (2016)CrossRefGoogle Scholar
  155. 155.
    Le Thi, H.A., Phan, D.N.: Efficient nonconvex group variable selection and application to group sparse optimal scoring (2017, Submitted)Google Scholar
  156. 156.
    Le Thi, H.A., Phan, D.N.: DC programming and DCA for sparse Fisher linear discriminant analysis. Neural Comput. Appl. 28(9), 2809–2822 (2017)CrossRefGoogle Scholar
  157. 157.
    Le Thi, H.A., Ta, A.S., Pham Dinh, T.: An efficient DCA based algorithm for power control in large scale wireless networks. Appl. Math. Comput. 318, 215–226 (2018)MathSciNetGoogle Scholar
  158. 158.
    Le Thi, H.A., Tran, D.Q.: Solving continuous min max problem for single period portfolio selection with discrete constraints by DCA. Optimization 61(8), 1025–1038 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  159. 159.
    Le Thi, H.A., Tran, D.Q.: New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains. J. Global Optim. 60(1), 5–24 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  160. 160.
    Le Thi, H.A., Tran, D.Q.: Optimizing a multi-stage production/inventory system by DC programming based approaches. Comput. Optim. Appl. 57(2), 441–468 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  161. 161.
    Le Thi, H.A., Tran, Q.D., Adjallah, K.H.: A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors. J. Ind. Manag. Optim. 10(1), 243–258 (2014)MathSciNetzbMATHGoogle Scholar
  162. 162.
    Le Thi, H.A., Tran, T.T., Pham Dinh, T., Gély, A.: DC programming and DCA for transmit beamforming and power allocation in multicasting relay network. In: Nguyen, T.B., van Do, T., Le Thi, H.A., Nguyen, N.T. (eds.) Advanced Computational Methods for Knowledge Engineering: ICCSAMA 2016, Proceedings, Part I, pp. 29–41. Springer, New York (2016)CrossRefGoogle Scholar
  163. 163.
    Le Thi, H.A., Vaz, A.I.F., Vicente, L.N.: Optimizing radial basis functions by D.C. programming and its use in direct search for global derivative-free optimization. TOP 20(1), 190–214 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  164. 164.
    Le Thi, H.A., Vo, X.T., Pham Dinh, T.: Feature selection for linear SVMs under uncertain data: robust optimization based on difference of convex functions algorithms. Neural Netw. 59, 36–50 (2014)zbMATHCrossRefGoogle Scholar
  165. 165.
    Le Thi, H.A., Vo, X.T., Pham Dinh, T.: Efficient nonegative matrix factorization by DC programming and DCA. Neural Comput. 28(6), 1163–1216 (2016)CrossRefGoogle Scholar
  166. 166.
  167. 167.
    Lee, J.D., Sun, Y., Saunders, M.A.: Proximal newton-type methods for minimizing composite functions. SIAM J. Optim. 24(3), 1420–1443 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  168. 168.
    de Leeuw, J.: Applications of convex analysis to multidimensional scaling. In: Barra, J.R., Brodeau, F., Romier, G., Van Cutsem, B. (eds.) Recent Developments in Statistics, pp. 133–146. North Holland, Amsterdam (1977)Google Scholar
  169. 169.
    Li, P., Rangapuram, S.S., Slawski, M.: Methods for sparse and low-rank recovery under simplex constraints. arXiv:1605.00507 (2016)
  170. 170.
    Li, Z., Lou, Y., Zeng, T.: Variational multiplicative noise removal by DC programming. J. Sci. Comput. 68(3), 1200–1216 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  171. 171.
    Liu, D., Shi, Y., Tian, Y., Huang, X.: Ramp loss least squares support vector machine. J. Comput. Sci. 14, 61–68 (2016)MathSciNetCrossRefGoogle Scholar
  172. 172.
    Liu, D., Tian, Y., Shi, Y.: Ramp loss nonparallel support vector machine for pattern classification. Knowl. Based Syst. 85, 224–233 (2015)CrossRefGoogle Scholar
  173. 173.
    Liu, Y., Shen, X.: Multicategory \(\psi \)-learning. J. Am. Stat. Assoc. 101, 500–509 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  174. 174.
    Liu, Y., Shen, X., Doss, H.: Multicategory \(\psi \)-learning and support vector machine: computational tools. J. Comput. Graph. Stat. 14, 219–236 (2005)MathSciNetCrossRefGoogle Scholar
  175. 175.
    Liu, Z.: Non-dominated set of a multi-objective optimisation problem. Ph.D. thesis, Lancaster University (2016)Google Scholar
  176. 176.
    Lou, Y., Osher, S., Xin, J.: Computational aspects of constrained L1–L2 minimization for compressive sensing. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N.T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 169–180. Springer, New York (2015)Google Scholar
  177. 177.
    Lou, Y., Yin, P., He, Q., Xin, J.: Computing sparse representation in a highly coherent dictionary based on difference of L1 and L2. J. Sci. Comput. 64(1), 178–196 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  178. 178.
    Lou, Y., Yin, P., Xin, J.: Point source super-resolution via non-convex \(l_1\) based methods. J. Sci. Comput. 68(3), 1082–1100 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  179. 179.
    Lou, Y., Zeng, T., Osher, S., Xin, J.: A weighted difference of anisotropic and isotropic total variation model for image processing. SIAM J. Imaging Sci. 8(3), 1798–1823 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  180. 180.
    Mahey, P., Phong, T.Q., Luna, H.P.L.: Separable convexification and DCA techniques for capacity and flow assignment problems. RAIRO Oper. Res. 35, 269–281 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  181. 181.
    Mangasarian, O.L.: Machine learning via polyhedral concave minimization. In: Fischer, H., Riedmueller, B., Schaeffler, S. (eds.) Applied Mathematics and Parallel Computing—Festschrift for Klaus Ritter, pp. 175–188. Physica-Verlag, Germany (1996)CrossRefGoogle Scholar
  182. 182.
    Martinet, B.: Brève communication. régularisation d’inéquations variationnelles par approximations successives. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 4(R3), 154–158 (1970)Google Scholar
  183. 183.
    Mokhtari, A., Koppel, A., Scutari, G., Ribeiro, A.: Large-scale nonconvex stochastic optimization by doubly stochastic successive convex approximation. In: 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4701–4705 (2017)Google Scholar
  184. 184.
    Mu, P., Hu, X., Wang, B., Li, Z.: Secrecy rate maximization with uncoordinated cooperative jamming by single-antenna helpers under secrecy outage probability constraint. IEEE Commun. Lett. 19(12), 2174–2177 (2015)CrossRefGoogle Scholar
  185. 185.
    Ndiaye, B.M.: Simulation et optimisation DC dans les réseaux de transport combinés : codes à usage industriel. Ph.D. thesis, INSA de Rouen (2007)Google Scholar
  186. 186.
    Ndiaye, B.M., Le Thi, H.A., Pham Dinh, T.: Single straddle carrier routing problem in port container terminals: Mathematical model and solving approaches. Int. J. Intell. Inf. Database Syst. 6(6), 532–554 (2012)zbMATHGoogle Scholar
  187. 187.
    Ndiaye, B.M., Le Thi, H.A., Pham Dinh, T., Niu, Y.: DC programming and DCA for large-scale two-dimensional packing problems. In: Intelligent Information and Database Systems. LNCS, vol. 7197, pp. 321–330. Springer, Berlin Heidelberg (2012)Google Scholar
  188. 188.
    Ndiaye, B.M., Pham Dinh, T., Le Thi, H.A.: Single straddle carrier routing problem in port container terminals: Mathematical model and solving approaches. In: Le Thi, H.A., Bouvry, P., Pham Dinh, T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 21–31 (2008)Google Scholar
  189. 189.
    Neumann, J., Schnörr, C., Steidl, G.: Combined SVM-based feature selection and classification. Mach. Learn. 61(1–3), 129–150 (2005)zbMATHCrossRefGoogle Scholar
  190. 190.
    Nguyen, D.M.: The DC programming and the cross- entropy method for some classes of problems in finance, assignment and search theory. Ph.D. thesis, INSA de Rouen (2012)Google Scholar
  191. 191.
    Nguyen, M.C.: La programmation DC et DCA pour certaines classes de problèmes en apprentissage et fouille de données. Ph.D. thesis, University of Lorraine (2014)Google Scholar
  192. 192.
    Nguyen, Q.T.: Approches locales et globales basées sur la programmation DC et DCA pour des problèmes combinatoires en variables mixtes 0-1 : applications à la planification opérationnelle. Ph.D. thesis, Université Paul Verlaine-Metz (2010)Google Scholar
  193. 193.
    Nguyen, Q.T., Le Thi, H.A.: Solving an inventory routing problem in supply chain by DC programming and DCA. In: Intelligent Information and Database Systems. LNCS, vol. 6592, pp. 432–441. Springer, Berlin Heidelberg (2011)Google Scholar
  194. 194.
    Nguyen, T.A., Nguyen, M.N.: Convergence analysis of a proximal point algorithm for minimizing differences of functions. Optimization 66(1), 129–147 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  195. 195.
    Nguyen, T.B.T.: La programmation DC et DCA en analyse d’image : acquisition comprimée, segmentation et restauration. Ph.D. thesis, University of Lorraine (2014)Google Scholar
  196. 196.
    Nguyen, T.B.T., Le Thi, H.A., Le, H.M., Vo, X.T.: DC approximation approach for \(\ell _0\)-minimization in compressed sensing. In: Le Thi, H.A., Nguyen, N.T., van Do, T. (eds.) Advanced Computational Methods for Knowledge Engineering, pp. 37–48. Springer, New York (2015)CrossRefGoogle Scholar
  197. 197.
    Nguyen, T.M.T., Le Thi, H.A.: A DC programming approach to the continuous equilibrium network design problem. In: Nguyen, T.B., van Do, T., Le Thi, H.A., Nguyen, N.T. (eds.) Advanced Computational Methods for Knowledge Engineering: ICCSAMA 2016, Proceedings, Part I, pp. 3–16. Springer, New York (2016)CrossRefGoogle Scholar
  198. 198.
    Nguyen, T.P.: Techniques d’optimisation en traitement d’image et vision par ordinateur et en transport logistique. Ph.D. thesis, Université Paul Verlaine-Metz (2007)Google Scholar
  199. 199.
    Nguyen, V.V.: Méthodes exactes pour l’optimisation DC polyédrale en variables mixtes 0-1 basées sur DCA et des nouvelles coupes. Ph.D. thesis, INSA de Rouen (2006)Google Scholar
  200. 200.
    Nguyen Canh, N., Le Thi, H.A., Pham Dinh, T.: A branch and bound algorithm based on DC programming and DCA for strategic capacity planning in supply chain design for a new market opportunity. In: Operations Research Proceedings. Operations Research Proceedings, vol. 2006, pp. 515–520. Springer, Berlin Heidelberg (2007)Google Scholar
  201. 201.
    Nguyen Canh, N., Pham, T.H., Tran, V.H.: DC programming and DCA approach for resource allocation optimization in OFDMA/TDD wireless networks. In: Le Thi, H.A., Nguyen, N.T., van Do, T. (eds.) Advanced Computational Methods for Knowledge Engineering, pp. 49–56. Springer, New York (2015)zbMATHCrossRefGoogle Scholar
  202. 202.
    Niu, Y.S., Júdice, J., Le Thi, H.A., Pham Dinh, T.: Solving the quadratic eigenvalue complementarity problem by DC programming. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N.T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 203–214. Springer, New York (2015)Google Scholar
  203. 203.
    Niu, Y.S., Pham Dinh, T., Le Thi, H.A., Judice, J.: Efficient DC programming approaches for asymmetric eigenvalue complementarity problem. Optim. Methods Softw. 28(4), 812–829 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  204. 204.
    Ong, C.S., Le Thi, H.A.: Learning sparse classifiers with difference of convex functions algorithms. Optim. Methods Softw. 28(4), 830–854 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  205. 205.
    Orlov, A., Strekalovsky, A.: On a local search for hexamatrix games. In: A. Kononov, I. Bykadorov, O. Khamisov, I. Davydov, P. Kononova (eds.) DOOR 2016, pp. 477–488 (2016)Google Scholar
  206. 206.
    Ortega, J., Rheinboldt, W.: Iterative Solutions of Nonlinear Equations in Several Variables, pp. 253–255. Academic, New York (1970)zbMATHGoogle Scholar
  207. 207.
    Pan, W., Shen, X., Liu, B.: Cluster analysis: unsupervised learning via supervised learning with a non-convex penalty. J. Mach. Learn. Res. 14(1), 1865–1889 (2013)MathSciNetzbMATHGoogle Scholar
  208. 208.
    Pang, J.S., Razaviyayn, M., Alvarado, A.: Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. 42(1), 95–118 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  209. 209.
    Pang, J.S., Tao, M.: Decomposition methods for computing directional stationary solutions of a class of non-smooth non-convex optimization problems. SIAM J. Optim. (2017, submitted)Google Scholar
  210. 210.
    Parida, P., Das, S.S.: Power allocation in OFDM based NOMA systems: a DC programming approach. In: 2014 IEEE Globecom Workshops (GC Wkshps), pp. 1026–1031. IEEE (2014)Google Scholar
  211. 211.
    Park, F., Lou, Y., Xin, J.: A weighted difference of anisotropic and isotropic total variation for relaxed Mumford-Shah image segmentation. In: IEEE ICIP 2016, pp. 4314–4318 (2016)Google Scholar
  212. 212.
    Park, S.H., Simeone, O., Sahin, O., Shamai, S.: Multihop backhaul compression for the uplink of cloud radio access networks. IEEE Trans. Veh. Technol. 65(5), 3185–3199 (2016)CrossRefGoogle Scholar
  213. 213.
    Pham, V.N.: Programmation DC et DCA pour l’optimisation non convexe/optimisation globale en variables mixtes entières : Codes et Applications. Ph.D. thesis, INSA de Rouen (2013)Google Scholar
  214. 214.
    Pham, V.N., Le Thi, H.A., Pham Dinh, T.: A DC programming framework for portfolio selection by minimizing the transaction costs. In: Advanced Computational Methods for Knowledge Engineering, Studies in Computational Intelligence, vol. 479, pp. 31–40. Springer International Publishing (2013)Google Scholar
  215. 215.
    Pham Dinh, T.: Elements homoduaux d’une matrice \(A\) relatifs à un couple de normes \((\phi ,\psi )\). Applications au calcul de \(s_{\phi \psi }(a)\) . Séminaire d’Analyse Numérique, Grenoble (1975)Google Scholar
  216. 216.
    Pham Dinh, T.: Calcul du maximum d’une forme quadratique définie positive sur la boule unité de la norme du maximum . Séminaire d’Analyse Numérique, Grenoble (1976)Google Scholar
  217. 217.
    Pham Dinh, T.: Contribution à la théorie de normes et ses applications à l’analyse numérique. Université Joseph Fourier, Grenoble, Thèse de doctorat d’etat es science (1981)Google Scholar
  218. 218.
    Pham Dinh, T.: Algorithmes de calcul du maximum des formes quadratiques sur la boule unité de la norme du maximum. Numer. Math. 45(3), 377–401 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  219. 219.
    Pham Dinh, T.: Convergence of a subgradient method for computing the bound norm of matrices. Linear Algebra Appl. 62, 163–182 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  220. 220.
    Pham Dinh, T., Ho, V.T., Le Thi, H.A.: DC programming and DCA for solving Brugnano-Casulli piecewise linear systems. Comput. Oper. Res. 87(Supplement C), 196–204 (2017)MathSciNetCrossRefGoogle Scholar
  221. 221.
    Pham Dinh, T., Le Thi, H.A.: Lagrangian stability and global optimality in nonconvex quadratic minimization over Euclidiean balls and spheres. J. Convex Anal. 2(1–2), 263–276 (1995)MathSciNetzbMATHGoogle Scholar
  222. 222.
    Pham Dinh, T., Le Thi, H.A.: Difference of convex function optimization algorithms (DCA) for globally minimizing nonconvex quadratic forms on Euclidean balls and spheres. Oper. Res. Lett. 19(5), 207–216 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  223. 223.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to D.C. programming: theory, algorithm and applications. Acta Math. Vietnam. 22(1), 289–355 (1997)MathSciNetzbMATHGoogle Scholar
  224. 224.
    Pham Dinh, T., Le Thi, H.A.: D.C. optimization algorithms for solving the trust region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  225. 225.
    Pham Dinh, T., Le Thi, H.A.: Recent advances in DC programming and DCA. In: Transactions on Computational Intelligence XIII. LNCS, vol. 8342, pp. 1–37. Springer, Berlin Heidelberg (2014)Google Scholar
  226. 226.
    Pham Dinh, T., Le Thi, H.A., Akoa, F.: Combining DCA and interior point techniques for large-scale nonconvex quadratic programming. Optim. Methods Softw. 23(4), 609–629 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  227. 227.
    Pham Dinh, T., Le Thi, H.A., Pham, V.N., Niu, Y.S.: DC programming approaches for discrete portfolio optimization under concave transaction costs. Optim. Lett. 10(2), 1–22 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  228. 228.
    Pham Dinh, T., Nguyen Canh, N., Le Thi, H.A.: DC programming and DCA for globally solving the value-at-risk. Comput. Manag. Sci. 6(4), 477–501 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  229. 229.
    Pham Dinh, T., Nguyen Canh, N., Le Thi, H.A.: An efficient combination of DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs. J. Global Optim. 48(4), 595–632 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  230. 230.
    Pham Dinh, T., Niu, Y.S.: An efficient DC programming approach for portfolio decision with higher moments. Comput. Optim. Appl. 50(3), 525–554 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  231. 231.
    Pham Dinh, T., Pham, V.N., Le Thi, H.A.: DC programming and DCA for portfolio optimization with linear and fixed transaction costs. In: Intelligent Information and Database Systems, LNCS, vol. 8398, pp. 392–402. Springer International Publishing (2014)Google Scholar
  232. 232.
    Pham Dinh, T., Souad, E.B.: Algorithms for solving a class of nonconvex optimization problems. Methods of subgradients. In: Hiriart-Urruty, J.B. (ed.) Fermat Days 85: Mathematics for Optimization, North-Holland Mathematics Studies, vol. 129, pp. 249–271. North-Holland, Amsterdam (1986)CrossRefGoogle Scholar
  233. 233.
    Pham Dinh, T., Souad, E.B.: Duality in D.C. (difference of convex functions) optimization. Subgradient methods. In: Trends in Mathematical Optimization, International Series of Numerical Mathematics, vol. 84, pp. 276–294. Birkhäuser, Basel (1988)Google Scholar
  234. 234.
    Phan, A.H., Tuan, H.D., Kha, H.H.: D.C. iterations for SINR maximin multicasting in cognitive radio. In: 6th International Conference on Signal Processing and Communication Systems (ICSPCS 2012), pp. 1–5 (2012)Google Scholar
  235. 235.
    Phan, D.N.: DCA based algorithms for learning with sparsity in high dimensional setting and stochastical learning. Ph.D. thesis, University of Lorraine (2016)Google Scholar
  236. 236.
    Phan, D.N., Le Thi, H.A., Pham Dinh, T.: Sparse covariance matrix estimation by DCA-based algorithms. Neural Comput. 29(11), 3040–3077 (2017)CrossRefGoogle Scholar
  237. 237.
    Piot, B., Geist, M., Pietquin, O.: Difference of convex functions programming for reinforcement learning. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 27, pp. 2519–2527. Curran Associates, Red Hook (2014)Google Scholar
  238. 238.
    Polyak, B.T.: Introduction to Optimization. Optimization Software. Inc. Publication Division, New York (1987)zbMATHGoogle Scholar
  239. 239.
    Poulakis, M.I., Vassaki, S., Panagopoulos, A.D.: Secure cooperative communications under secrecy outage constraint: a DC programming approach. IEEE Wirel. Commun. Lett. 5(3), 332–335 (2016)CrossRefGoogle Scholar
  240. 240.
    Queiroz, M., Júdice, J., Humes, C.: The symmetric eigenvalue complementarity problem. Math. Comput. 73(248), 1849–1863 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  241. 241.
    Rakotomamonjy, A., Flamary, R., Gasso, G.: DC proximal newton for nonconvex optimization problems. IEEE Trans. Neural Netw. Learn. Syst. 27(3), 636–647 (2016)MathSciNetCrossRefGoogle Scholar
  242. 242.
    Razaviyayn, M.: Successive convex approximation: analysis and applications. Ph.D. thesis, University of Minnesota (2014)Google Scholar
  243. 243.
    Razaviyayn, M., Hong, M., Luo, Z.Q.: A unified convergence analysis of block successive minimization methods for nonsmooth optimization. SIAM J. Optim. 23(2), 1126–1153 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  244. 244.
    Razaviyayn, M., Hong, M., Luo, Z.Q., Pang, J.S.: Parallel successive convex approximation for nonsmooth nonconvex optimization. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N., Weinberger, K. (eds.) Advances in Neural Information Processing Systems, vol. 27, pp. 1440–1448. Curran Associates, Red Hook (2014)Google Scholar
  245. 245.
    Razaviyayn, M., Sanjabi, M., Luo, Z.Q.: A stochastic successive minimization method for nonsmooth nonconvex optimization with applications to transceiver design in wireless communication networks. Math. Program. 157(2), 515–545 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  246. 246.
    Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22(3), 400–407 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  247. 247.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHCrossRefGoogle Scholar
  248. 248.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  249. 249.
    Schad, A., Law, K.L., Pesavento, M.: Rank-two beamforming and power allocation in multicasting relay networks. IEEE Trans Signal Process. 63(13), 3435–3447 (2015)MathSciNetCrossRefGoogle Scholar
  250. 250.
    Schleich, J., Le Thi, H.A., Bouvry, P.: Solving the minimum \(m\)-dominating set problem by a continuous optimization approach based on DC programming and DCA. J. Comb. Optim. 24(4), 397–412 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  251. 251.
    Schnörr, C.: Signal and image approximation with level-set constraints. Computing 81(2), 137–160 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  252. 252.
    Schüle, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discrete Appl. Math. 151(1–3), 229–243 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  253. 253.
    Schüle, T., Weber, S., Schnörr, C.: Adaptive reconstruction of discrete-valued objects from few projections. Electron. Notes Discrete Math. 20, 365–384 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  254. 254.
    Scutari, G., Facchinei, F., Lampariello, L.: Parallel and distributed methods for constrained nonconvex optimization-part I: theory. IEEE Trans. Signal Process. 65(8), 1929–1944 (2017)MathSciNetCrossRefGoogle Scholar
  255. 255.
    Scutari, G., Facchinei, F., Lampariello, L., Sardellitti, S., Song, P.: Parallel and distributed methods for constrained nonconvex optimization-part II: applications in communications and machine learning. IEEE Trans. Signal Process. 65(8), 1945–1960 (2017)MathSciNetCrossRefGoogle Scholar
  256. 256.
    Scutari, G., Facchinei, F., Song, P., Palomar, D.P., Pang, J.S.: Decomposition by partial linearization: parallel optimization of multi-agent systems. IEEE Trans. Signal Process. 62(3), 641–656 (2014)MathSciNetCrossRefGoogle Scholar
  257. 257.
    Seeger, A.: Quadratic eigenvalue problems under conic constraints. SIAM J. Matrix Anal. A 32(3), 700–721 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  258. 258.
    Shen, X., Huang, H.C.: Simultaneous supervised clustering and feature selection over a graph. Biometrika 99(4), 899–914 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  259. 259.
    Shen, X., Tseng, G.C., Zhang, X., Wong, W.H.: On \(\psi \) learning. J. Am. Stat. Assoc. 98, 724–734 (2003)zbMATHCrossRefGoogle Scholar
  260. 260.
    Slawski, M., Hein, M., Lutsik, P.: Matrix factorization with binary components. In: Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 26, pp. 3210–3218. Curran Associates, Red Hook (2013)Google Scholar
  261. 261.
    Smola, A.J., Song, L., Teo, C.H.: Relative novelty detection. In: Proceedings of the 12th International Conference on Artificial Intelligence and Statistics, vol. 5. JMLR W&CP 5, pp. 536–543 (2009)Google Scholar
  262. 262.
    Song, Y., Lin, L., Jian, L.: Robust check loss-based variable selection of high-dimensional single-index varying-coefficient model. Commun. Nonlinear Sci. 36, 109–128 (2016)MathSciNetCrossRefGoogle Scholar
  263. 263.
    Sriperumbudur, B.K., Torres, D.A., Lanckriet, G.R.G.: Sparse eigen methods by D.C. programming. In: ICML’07, pp. 831–838. ACM, New York, NY, USA (2007)Google Scholar
  264. 264.
    Sun, Q., Xiang, S., Ye, J.: Robust principal component analysis via capped norms. In: Proceedings of the 19th ACM SIGKDD, KDD’13, pp. 311–319. ACM (2013)Google Scholar
  265. 265.
    Sun, W., Sampaio, J.B., Candido, R.M.: Proximal point algorithm for minimization of DC function. J. Comput. Math. 21, 451–462 (2003)MathSciNetzbMATHGoogle Scholar
  266. 266.
    Ta, A.S.: Programmation DC et DCA pour la résolution de certaines classes des problèmes dans les systèmes de transport et de communication. Ph.D. thesis, INSA - Rouen (2012)Google Scholar
  267. 267.
    Ta, A.S., Le Thi, H.A., Arnould, G., Khadraoui, D., Pham Dinh, T.: Solving car pooling problem using DCA. In: Global Information Infrastructure Symposium (GIIS 2011), pp. 1–6 (2011)Google Scholar
  268. 268.
    Ta, A.S., Le Thi, H.A., Ha, T.S.: Solving relaxation orienteering problem using DCA-CUT. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N.T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 191–202. Springer, New York (2015)Google Scholar
  269. 269.
    Ta, A.S., Le Thi, H.A., Khadraoui, D., Pham Dinh, T.: Solving multicast QoS routing problem in the context V2I communication services using DCA. In: IEEE/ACIS 9th International Conference on Computer and Information Science (ICIS), 2010, pp. 471–476 (2010)Google Scholar
  270. 270.
    Ta, A.S., Le Thi, H.A., Khadraoui, D., Pham Dinh, T.: Solving QoS routing problems by DCA. In: Intelligent Information and Database Systems. LNCS, vol. 5991, pp. 460–470. Springer, Berlin Heidelberg (2010)Google Scholar
  271. 271.
    Ta, A.S., Le Thi, H.A., Khadraoui, D., Pham Dinh, T.: Solving partitioning-hub location-routing problem using DCA. J. Ind. Manag. Optim. 8(1), 87–102 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  272. 272.
    Ta, A.S., Pham Dinh, T., Le Thi, H.A., Khadraoui, D.: Solving many to many multicast QoS routing problem using DCA and proximal decomposition technique. In: International Conference on Computing, Networking and Communications (ICNC 2012), pp. 809–814 (2012)Google Scholar
  273. 273.
    Ta, M.T.: Techniques d’optimisation non convexe basée sur la programmation DC et DCA et méthodes évolutives pour la classification non supervisée. Ph.D. thesis, University of Lorraine (2014)Google Scholar
  274. 274.
    Ta, M.T., Le Thi, H.A., Boudjeloud-Assala, L.: Clustering data stream by a sub-window approach using DCA. In: Perner, P. (ed.) Machine Learning and Data Mining in Pattern Recognition, pp. 279–292. Springer, Berlin (2012)CrossRefGoogle Scholar
  275. 275.
    Ta, M.T., Le Thi, H.A., Boudjeloud-Assala, L.: Clustering data streams over sliding windows by DCA. In: Nguyen, T.N., van Do, T., le Thi, A.H. (eds.) Advanced Computational Methods for Knowledge Engineering, pp. 65–75. Springer, Heidelberg (2013)Google Scholar
  276. 276.
    Ta, M.T., Le Thi, H.A., Boudjeloud-Assala, L.: An efficient clustering method for massive dataset based on DC programming and DCA approach. In: Lee, M., Hirose, A., Hou, Z.G., Kil, R.M. (eds.) ICONIP 2013, Part II, LNCS, vol. 8227, pp. 538–545. Springer, Berlin Heidelberg (2013)Google Scholar
  277. 277.
    Taleb, D., Liu, Y., Pesavento, M.: Full-rate general rank beamforming in single-group multicasting networks using non-orthogonal STBC. In: 24th EUSIPCO, pp. 2365–2369 (2016)Google Scholar
  278. 278.
    Thai, J., Hunter, T., Akametalu, A.K., Tomlin, C.J., Bayen, A.M.: Inverse covariance estimation from data with missing values using the concave-convex procedure. In: 53rd IEEE Conference on Decision and Control, pp. 5736–5742 (2014)Google Scholar
  279. 279.
    Thanh, P.N., Bostel, N., Péton, O.: A DC programming heuristic applied to the logistics network design problem. Int. J. Prod. Econ. 135(1), 94–105 (2012)CrossRefGoogle Scholar
  280. 280.
    Thiao, M.: Pham Dinh, T., Le Thi, H.A.: DC programming approach for a class of nonconvex programs involving \(\ell _0\) norm. Modelling. In: Computation and Optimization in Information Systems and Management Sciences, Communications in Computer and Information Science, vol. 14, pp. 348–357. Springer, Berlin Heidelberg (2008)Google Scholar
  281. 281.
    Thiao, M., Pham Dinh, T., Le Thi, H.A.: A DC programming approach for sparse eigenvalue problem. In: Fürnkranz, J., Joachims, T. (eds.) Proceedings ICML-10, pp. 1063–1070. Omnipress (2010)Google Scholar
  282. 282.
    Tian, X., Gasso, G., Canu, S.: A multiple kernel framework for inductive semi-supervised SVM learning. Neurocomputing 90, 46–58 (2012)CrossRefGoogle Scholar
  283. 283.
    Torres, D.A., Turnbull, D., Sriperumbudur, B.K., Barrington, L., Lanckriet, G.R.G.: Finding musically meaningful words by sparse CCA. In: NIPS Workshop on Music, the Brain and Cognition (2007)Google Scholar
  284. 284.
    Tran, D.Q., Le Thi, H.A., Adjallah, K.H.: DCA for minimizing the cost and tardiness of preventive maintenance tasks under real-time allocation constraint. In: Nguyen, N.T., Le, M.T., Swiatek, J. (eds.) Intelligent Information and Database Systems, LNCS, vol. 5991, pp. 410–419. Springer, Berlin Heidelberg (2010)Google Scholar
  285. 285.
    Tran, D.Q., Nguyen, B.T.P., Nguyen, Q.T.: A new approach for optimizing traffic signals in networks considering rerouting. In: Modelling, Computation and Optimization in Information Systems and Management Sciences, Advances in Intelligent Systems and Computing, vol. 359, pp. 143–154. Springer International Publishing (2015)Google Scholar
  286. 286.
    Tran, T.T., Le Thi, H.A., Pham Dinh, T.: DC programming and DCA for a novel resource allocation problem in emerging area of cooperative physical layer security. In: Advanced Computational Methods for Knowledge Engineering, Advances in Intelligent Systems and Computing 358, 57–68 (2015)Google Scholar
  287. 287.
    Tran, T.T., Le Thi, H.A., Pham Dinh, T.: DC programming and DCA for enhancing physical layer security via cooperative jamming. Comput. Oper. Res. 87(Supplement C), 235–244 (2017)MathSciNetCrossRefGoogle Scholar
  288. 288.
    Tran, T.T., Tuan, N.N., Le Thi, H.A., Gély, A.: DC programming and DCA for enhancing physical layer security via relay beamforming strategies. In: Nguyen, N.T., Trawiński, B., Fujita, H., Hong, T.P. (eds.) ACIIDS 2016, Part II, LNAI 9622, pp. 640–650. Springer, Berlin Heidelberg (2016)Google Scholar
  289. 289.
    Tsiligkaridis, T., Marcheret, E., Goel, V.: A difference of convex functions approach to large-scale log-linear model estimation. IEEE Trans. Audio Speech 21(11), 2255–2266 (2013)CrossRefGoogle Scholar
  290. 290.
    Tuan, H.N.: Convergence rate of the Pham Dinh-Le Thi algorithm for the trust-region subproblem. J. Optim. Theory Appl. 154(3), 904–915 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  291. 291.
    Tuan, H.N., Yen, N.D.: Convergence of Pham Dinh-Le Thi’s algorithm for the trust-region subproblem. J. Global Optim. 55(2), 337–347 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  292. 292.
    Vanderbei, R.J.: LOQO: an interior point code for quadratic programming. Optim. Methods Softw. 11(1–4), 451–484 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  293. 293.
    Vasiloglou, N., Gray, A.G., Anderson, D.V.: Non-negative matrix factorization, convexity and isometry. In: Proceedings of the 2009 SIAM ICDM, chap. 57, pp. 673–684 (2009)Google Scholar
  294. 294.
    Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, Oxford (1991)zbMATHGoogle Scholar
  295. 295.
    Vo, X.T.: Learning with sparsity and uncertainty by difference of convex functions optimization. Ph.D. thesis, University of Lorraine (2015)Google Scholar
  296. 296.
    Vo, X.T., Le Thi, H.A.: Pham Dinh, T.: Robust optimization for clustering. ACIIDS 2016. Part II, LNCS, vol. 9622, pp. 1–10. Springer, Berlin Heidelberg (2016)Google Scholar
  297. 297.
    Vo, X.T., Le Thi, H.A., Pham Dinh, T., Nguyen, T.B.T.: DC programming and DCA for dictionary learning. In: Computational Collective Intelligence, LNCS, vol. 9329, pp. 295–304. Springer International Publishing (2015)Google Scholar
  298. 298.
    Vo, X.T., Tran, B., Le Thi, H.A., Pham Dinh, T.: Ramp loss support vector data description. In: Proc. 9th Asian Conference on Intelligent Information and Database Systems (ACIIDS 2017). 3–5 April 2017, Kanazawa, Japan (2017). Lecture Note in Computer Science. Springer (2017, to appear)Google Scholar
  299. 299.
    Vucic, N., Shi, S., Schubert, M.: DC programming approach for resource allocation in wireless networks. In: Proceedings of the 8th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt 2010), pp. 380–386 (2010)Google Scholar
  300. 300.
    Wang, D., Chen, W., Han, Z.: Energy efficient secure communication over decode-and-forward relay channels. IEEE Trans. Commun. 63(3), 892–905 (2015)CrossRefGoogle Scholar
  301. 301.
    Wang, F., Zhao, B., Zhang, C.: Linear time maximum margin clustering. IEEE Trans. Neural Netw. 21(2), 319–332 (2010)CrossRefGoogle Scholar
  302. 302.
    Wang, J., Shen, X.: Large margin semi-supervised learning. J. Mach. Learn. Res. 8, 1867–1891 (2007)MathSciNetzbMATHGoogle Scholar
  303. 303.
    Wang, J., Shen, X., Pan, W.: On transductive support vector machines. In: Prediction and Discovery, Contemporary Mathematics 443, pp. 7–19. American Mathematical Society (2007)Google Scholar
  304. 304.
    Wang, J., Shen, X., Pan, W.: On efficient large margin semisupervised learning: method and theory. J. Mach. Learn. Res. 10, 719–742 (2009)MathSciNetzbMATHGoogle Scholar
  305. 305.
    Wang, K., Zhong, P., Zhao, Y.: Training robust support vector regression via D.C. program. J. Inf. Comput. Sci. 7(12), 2385–2394 (2010)Google Scholar
  306. 306.
    Wang, K., Zhu, W., Zhong, P.: Robust support vector regression with generalized loss function and applications. Neural Process. Lett. 41(1), 89–106 (2015)CrossRefGoogle Scholar
  307. 307.
    Wang, L., Kim, Y., Li, R.: Calibrating nonconvex penalized regression in ultra-high dimension. Ann. Stat. 41(5), 2505–2536 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  308. 308.
    Wang, Y., Xia, X.: An effective \(l_0\)-svm classifier for face recognition based on haar features. Adv. Nat. Sci. 9(1), 1–4 (2016)Google Scholar
  309. 309.
    Weber, S., Schüle, T., Schnörr, C.: Prior learning and convex–concave regularization of binary tomography. Electron. Notes Discrete Math. 20, 313–327 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  310. 310.
    Weston, J., Elisseeff, A., Schölkopf, B., Tipping, M.: Use of the zero-norm with linear models and kernel methods. J. Mach. Learn. Res. 3, 1439–1461 (2003)MathSciNetzbMATHGoogle Scholar
  311. 311.
    Wozabal, D.: Value-at-risk optimization using the difference of convex algorithm. OR Spectrum 34(4), 861–883 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  312. 312.
    Wu, C., Kwon, S., Shen, X., Pan, W.: A new algorithm and theory for penalized regression-based clustering. J. Mach. Learn. Res. 17, 1–25 (2016)MathSciNetzbMATHGoogle Scholar
  313. 313.
    Wu, C., Li, C., Long, Q.: A DC Programming approach for sensor network localization with uncertainties in anchor positions. J. Ind. Manag. Optim. 10(3), 817–826 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  314. 314.
    Wu, Y., Liu, Y.: Robust truncated hinge loss support vector machines. J. Am. Stat. Assoc. 102(479), 974–983 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  315. 315.
    Wu, Y., Liu, Y.: Variable selection in quantile regression. Stat Sin. 19, 801–817 (2009)MathSciNetzbMATHGoogle Scholar
  316. 316.
    Xiang, S., Shen, X., Ye, J.: Efficient nonconvex sparse group feature selection via continuous and discrete optimization. Artif. Intell. 224, 28–50 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  317. 317.
    Yang, L., Ju, R.: A DC programming approach for feature selection in the minimax probability machine. Int. J. Comput. Intell. Syst. 7(1), 12–24 (2014)CrossRefGoogle Scholar
  318. 318.
    Yang, L., Qian, Y.: A sparse logistic regression framework by difference of convex functions programming. Appl. Intell. 45(2), 241–254 (2016)CrossRefGoogle Scholar
  319. 319.
    Yang, L., Wang, L.: A class of semi-supervised support vector machines by DC programming. Adv. Data Anal. Classif. 7(4), 417–433 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  320. 320.
    Yang, L., Zhang, S.: A sparse extreme learning machine framework by continuous optimization algorithms and its application in pattern recognition. Eng. Appl. Artif. Int. 53, 176–189 (2016)CrossRefGoogle Scholar
  321. 321.
    Yang, S., Yuan, L., Lai, Y.C., Shen, X., Wonka, P., Ye, J.: Feature grouping and selection over an undirected graph. In: ACM SIGKDD, pp. 922–930 (2012)Google Scholar
  322. 322.
    Yang, T., Liu, J., Gong, P., Zhang, R., Shen, X., Ye, J.: Absolute fused lasso and its application to genome-wide association studies. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’16, pp. 1955–1964. ACM (2016)Google Scholar
  323. 323.
    Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of \(\ell _{1-2}\) for compressed sensing. SIAM J. Sci. Comput. 37(1), 536–563 (2015)MathSciNetCrossRefGoogle Scholar
  324. 324.
    Yin, P., Xin, J., Qi, Y.: Linear feature transform and enhancement of classification on deep neural network. (2016, Submitted)Google Scholar
  325. 325.
    Ying, Y., Huang, K., Campbell, C.: Enhanced protein fold recognition through a novel data integration approach. BMC Bioinform. 10(1), 1–18 (2009)CrossRefGoogle Scholar
  326. 326.
    You, S., Lijun, C., Liu, Y.E.: Convex-concave procedure for weighted sum-rate maximization in a MIMO interference network. In: IEEE GLOBECOM 2014, pp. 4060–4065 (2014)Google Scholar
  327. 327.
    Yu, C.N.J., Joachims, T.: Learning structural SVMs with latent variables. In: ICML’09, pp. 1169–1176. ACM, New York, NY, USA (2009)Google Scholar
  328. 328.
    Yu, P.L.: Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions. In: Mathematical Concepts and Methods in Science and Engineering, vol. 30. Springer, USA (1985)Google Scholar
  329. 329.
    Yuille, A.L., Rangarajan, A.: The concave–convex procedure. Neural Comput. 15(4), 915–936 (2003)zbMATHCrossRefGoogle Scholar
  330. 330.
    Zhang, K., Tsang, I.W., Kwok, J.T.: Maximum margin clustering made practical. IEEE Trans. Neural Netw. 20(4), 583–596 (2009)CrossRefGoogle Scholar
  331. 331.
    Zhang, P., Tian, Y., Zhang, Z., Li, A., Zhu, X.: Select objective functions for multiple criteria programming classification. In: Web Intelligence and Intelligent Agent Technology, 2008. WI-IAT’08. IEEE/WIC/ACM International Conference on, vol. 3, pp. 420–423 (2008)Google Scholar
  332. 332.
    Zhang, X., Wu, Y., Wang, L., Li, R.: Variable selection for support vector machines in moderately high dimensions. J. R. Stat. Soc. B 78(1), 53–76 (2016)MathSciNetCrossRefGoogle Scholar
  333. 333.
    Zhao, Z., Sun, L., Yu, S., Liu, H., Ye, J.: Multiclass probabilistic kernel discriminant analysis. In: Proceedings of the 21st International Joint Conference on Artifical Intelligence, IJCAI’09, pp. 1363–1368. Morgan Kaufmann (2009)Google Scholar
  334. 334.
    Zheng, G.: Joint beamforming optimization and power control for full-duplex MIMO two-way relay channel. IEEE Trans. Signal Process. 63(3), 555–566 (2015)MathSciNetCrossRefGoogle Scholar
  335. 335.
    Zheng, G., Krikidis, I., Li, J., Petropulu, A.P., Ottersten, B.: Improving physical layer secrecy using full-duplex jamming receivers. IEEE Trans. Signal Process. 61(20), 4962–4974 (2013)MathSciNetCrossRefGoogle Scholar
  336. 336.
    Zhong, P.: Training robust support vector regression with smooth non-convex loss function. Optim. Methods Softw. 27(6), 1039–1058 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  337. 337.
    Zhong, Y., Aghezzaf, E.H.: Combining DC-programming and steepest-descent to solve the single-vehicle inventory routing problem. Comput. Ind. Eng. 61(2), 313–321 (2011)CrossRefGoogle Scholar
  338. 338.
    Zhou, Y., Zhu, Y., Xue, Z.: Enhanced MIMOME wiretap channel via adopting full-duplex MIMO radios. In: 2014 IEEE Global Communications Conference, pp. 3320–3325. IEEE (2014)Google Scholar
  339. 339.
    Zhou, Z.H., Zhang, M.L., Huang, S.J., Li, Y.F.: Multi-instance multi-label learning. Artif. Intell. 176(1), 2291–2320 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  340. 340.
    Zhu, Y., Shen, X., Pan, W.: Simultaneous grouping pursuit and feature selection over an undirected graph. J. Am. Stat. Assoc. 108(502), 713–725 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  341. 341.
    Zisler, M., Petra, S., Schnörr, C., Schnörr, C.: Discrete tomography by continuous multilabeling subject to projection constraints. In: Proceedings of the 38th German Conference on Pattern Recognition (2016)Google Scholar
  342. 342.
    Zou, H.: The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 2006(476), 1418–1429 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  343. 343.
    Zou, H., Li, R.: One-step sparse estimates in nonconcave penalized likelihood models. Ann. Stat. 36(4), 1509–1533 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Laboratory of Theoretical and Applied Computer Science (LITA)University of LorraineMetz TechnopoleFrance
  2. 2.Laboratory of MathematicsNational Institute for Applied Sciences - RouenSaint-Étienne-du-Rouvray CedexFrance

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