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DC programming and DCA: thirty years of developments

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  6. Arthanari, T.S., Le Thi, H.A.: New formulations of the multiple sequence alignment problem. Optim. Lett. 5(1), 27–40 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Astorino, A., Fuduli, A.: Semisupervised spherical separation. Appl. Math. Model. 39(20), 6351–6358 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Astorino, A., Fuduli, A., Gaudioso, M.: DC models for spherical separation. J. Global Optim. 48(4), 657–669 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Astorino, A., Fuduli, A., Gaudioso, M.: Margin maximization in spherical separation. Comput. Optim. Appl. 53(2), 301–322 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bačák, M., Borwein, J.M.: On difference convexity of locally lipschitz functions. Optimization 60(8–9), 961–978 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  15. Bertsekas, D.P.: Incremental proximal methods for large scale convex optimization. Math. Program. 129(2), 163–195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bien, J., Tibshirani, R.J.: Sparse estimation of a covariance matrix. Biometrika 98(4), 807–820 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  18. Bouallagui, S.: Techniques d’optimisation déterministe et stochastique pour la résolution de problèmes difficiles en cryptologie. Ph.D. thesis, INSA de Rouen (2010)

  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)

  20. Bradley, P.S., Mangasarian, O.L.: Feature selection via concave minimization and support vector machines. ICML 1998, 82–90 (1998)

    Google Scholar 

  21. Candes, E.J., Wakin, M., Boyd, S.: Enhancing sparsity by reweighted-\(l_{1}\) minimization. J. Fourier Anal. Appl. 14, 877–905 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  25. Chen, G., Zeng, D., Kosorok, M.R.: Personalized dose finding using outcome weighted learning. J. Am. Stat. Assoc. 111(516), 1509–1521 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  28. Collobert, R., Sinz, F., Weston, J., Bottou, L.: Large scale transductive SVMs. J. Mach. Learn. Res. 7, 1687–1712 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Collobert, R., Sinz, F., Weston, J., Bottou, L.: Trading convexity for scalability. In: ICML 2006, pp. 201–208 (2006)

  30. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Conn, A., Gould, N., Toint, P.: Trust Region Methods. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  35. Ellis, S.E., Nayakkankuppam, M.V.: Phylogenetic analysis via DC programming . (Preprint) (2003)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fastrich, B., Paterlini, S., Winker, P.: Constructing optimal sparse portfolios using regularization methods. CMS 12(3), 417–434 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  49. Guan, G., Gray, A.: Sparse high-dimensional fractional-norm support vector machine via DC programming. Comput. Stat. Data Anal. 67, 136–148 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  52. Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math. 9(3), 707–713 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  53. Heinkenschloss, M.: On the solution of a two ball trust region subproblem. Math. Program. 64(1–3), 249–276 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  55. Ho, V.T.: Advanced machine learning techniques based on DC programming and DCA. Ph.D. thesis, University of Lorraine (2017)

  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)

    Chapter  Google Scholar 

  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. 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)

    Article  Google Scholar 

  59. Huang, X., Shi, L., Suykens, J.: Ramp loss linear programming support vector machine. J. Mach. Learn. Res. 15(1), 2185–2211 (2014)

    MathSciNet  MATH  Google Scholar 

  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. IBM: CPLEX Optimizer. https://www.ibm.com/analytics/data-science/prescriptive-analytics/cplex-optimizer

  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)

  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)

    Article  Google Scholar 

  64. Júdice, J.J., Sherali, H.D., Ribeiro, I.M.: The eigenvalue complementarity problem. Comput. Optim. Appl. 37(2), 139–156 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Global Optim. 13(4), 389–406 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  68. Kim, S., Pan, W., Shen, X.: Network-based penalized regression with application to genomic data. Biometrics 69(3), 582–93 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  86. Le Thi, H.A.: DCA collaborative for clustering. University of Lorraine, Tech. rep. (2013)

  87. Le Thi, H.A.: Phylogenetic tree reconstruction by a DCA based algorithm. Tech. rep., LITA, University of Lorraine (2013)

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  90. Le Thi, H.A., Ho, V.T.: Online learning based on Online DCA (2016, Submitted)

  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)

    Chapter  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  93. Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: Convergence analysis of DCA with subanalytic data. J. Optim. Theory Appl. (2018)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Chapter  MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  147. Le Thi, H.A., Pham Dinh, T., Muu, L.D.: Exact penalty in DC programming. Vietnam J. Math. 27(2), 169–179 (1999)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  154. Le Thi, H.A., Phan, D.N.: DC programming and DCA for sparse optimal scoring problem. Neurocomputing 186, 170–181 (2016)

    Article  Google Scholar 

  155. Le Thi, H.A., Phan, D.N.: Efficient nonconvex group variable selection and application to group sparse optimal scoring (2017, Submitted)

  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)

    Article  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  166. Le Thi, H.A.: DC Programming and DCA: http://www.lita.univ-lorraine.fr/~lethi/index.php/en/research/dc-programming-and-dca.html (Homepage) (2005)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Li, P., Rangapuram, S.S., Slawski, M.: Methods for sparse and low-rank recovery under simplex constraints. arXiv:1605.00507 (2016)

  170. Li, Z., Lou, Y., Zeng, T.: Variational multiplicative noise removal by DC programming. J. Sci. Comput. 68(3), 1200–1216 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  171. Liu, D., Shi, Y., Tian, Y., Huang, X.: Ramp loss least squares support vector machine. J. Comput. Sci. 14, 61–68 (2016)

    Article  MathSciNet  Google Scholar 

  172. Liu, D., Tian, Y., Shi, Y.: Ramp loss nonparallel support vector machine for pattern classification. Knowl. Based Syst. 85, 224–233 (2015)

    Article  Google Scholar 

  173. Liu, Y., Shen, X.: Multicategory \(\psi \)-learning. J. Am. Stat. Assoc. 101, 500–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  174. Liu, Y., Shen, X., Doss, H.: Multicategory \(\psi \)-learning and support vector machine: computational tools. J. Comput. Graph. Stat. 14, 219–236 (2005)

    Article  MathSciNet  Google Scholar 

  175. Liu, Z.: Non-dominated set of a multi-objective optimisation problem. Ph.D. thesis, Lancaster University (2016)

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  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)

  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)

    MATH  Google Scholar 

  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)

  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)

  189. Neumann, J., Schnörr, C., Steidl, G.: Combined SVM-based feature selection and classification. Mach. Learn. 61(1–3), 129–150 (2005)

    Article  MATH  Google Scholar 

  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)

  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)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

  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)

  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)

  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)

    Chapter  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  206. Ortega, J., Rheinboldt, W.: Iterative Solutions of Nonlinear Equations in Several Variables, pp. 253–255. Academic, New York (1970)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  208. Pang, J.S., Razaviyayn, M., Alvarado, A.: Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. 42(1), 95–118 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

    Article  Google Scholar 

  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)

  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)

  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)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  219. Pham Dinh, T.: Convergence of a subgradient method for computing the bound norm of matrices. Linear Algebra Appl. 62, 163–182 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Chapter  Google Scholar 

  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)

  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)

  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)

  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)

    Article  Google Scholar 

  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. Polyak, B.T.: Introduction to Optimization. Optimization Software. Inc. Publication Division, New York (1987)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  240. Queiroz, M., Júdice, J., Humes, C.: The symmetric eigenvalue complementarity problem. Math. Comput. 73(248), 1849–1863 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  242. Razaviyayn, M.: Successive convex approximation: analysis and applications. Ph.D. thesis, University of Minnesota (2014)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  246. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22(3), 400–407 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  247. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  248. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  251. Schnörr, C.: Signal and image approximation with level-set constraints. Computing 81(2), 137–160 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  253. Schüle, T., Weber, S., Schnörr, C.: Adaptive reconstruction of discrete-valued objects from few projections. Electron. Notes Discrete Math. 20, 365–384 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  257. Seeger, A.: Quadratic eigenvalue problems under conic constraints. SIAM J. Matrix Anal. A 32(3), 700–721 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  258. Shen, X., Huang, H.C.: Simultaneous supervised clustering and feature selection over a graph. Biometrika 99(4), 899–914 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  259. Shen, X., Tseng, G.C., Zhang, X., Wong, W.H.: On \(\psi \) learning. J. Am. Stat. Assoc. 98, 724–734 (2003)

    Article  MATH  Google Scholar 

  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. 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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  265. Sun, W., Sampaio, J.B., Candido, R.M.: Proximal point algorithm for minimization of DC function. J. Comput. Math. 21, 451–462 (2003)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  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. 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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

    Chapter  Google Scholar 

  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. 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)

  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)

  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)

  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)

    Article  Google Scholar 

  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)

  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)

  282. Tian, X., Gasso, G., Canu, S.: A multiple kernel framework for inductive semi-supervised SVM learning. Neurocomputing 90, 46–58 (2012)

    Article  Google Scholar 

  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)

  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)

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  292. Vanderbei, R.J.: LOQO: an interior point code for quadratic programming. Optim. Methods Softw. 11(1–4), 451–484 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  294. Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, Oxford (1991)

    MATH  Google Scholar 

  295. Vo, X.T.: Learning with sparsity and uncertainty by difference of convex functions optimization. Ph.D. thesis, University of Lorraine (2015)

  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)

  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)

  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)

  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)

  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)

    Article  Google Scholar 

  301. Wang, F., Zhao, B., Zhang, C.: Linear time maximum margin clustering. IEEE Trans. Neural Netw. 21(2), 319–332 (2010)

    Article  Google Scholar 

  302. Wang, J., Shen, X.: Large margin semi-supervised learning. J. Mach. Learn. Res. 8, 1867–1891 (2007)

    MathSciNet  MATH  Google Scholar 

  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)

  304. Wang, J., Shen, X., Pan, W.: On efficient large margin semisupervised learning: method and theory. J. Mach. Learn. Res. 10, 719–742 (2009)

    MathSciNet  MATH  Google Scholar 

  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. Wang, K., Zhu, W., Zhong, P.: Robust support vector regression with generalized loss function and applications. Neural Process. Lett. 41(1), 89–106 (2015)

    Article  Google Scholar 

  307. Wang, L., Kim, Y., Li, R.: Calibrating nonconvex penalized regression in ultra-high dimension. Ann. Stat. 41(5), 2505–2536 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  309. Weber, S., Schüle, T., Schnörr, C.: Prior learning and convex–concave regularization of binary tomography. Electron. Notes Discrete Math. 20, 313–327 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  311. Wozabal, D.: Value-at-risk optimization using the difference of convex algorithm. OR Spectrum 34(4), 861–883 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  314. Wu, Y., Liu, Y.: Robust truncated hinge loss support vector machines. J. Am. Stat. Assoc. 102(479), 974–983 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  315. Wu, Y., Liu, Y.: Variable selection in quantile regression. Stat Sin. 19, 801–817 (2009)

    MathSciNet  MATH  Google Scholar 

  316. Xiang, S., Shen, X., Ye, J.: Efficient nonconvex sparse group feature selection via continuous and discrete optimization. Artif. Intell. 224, 28–50 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  318. Yang, L., Qian, Y.: A sparse logistic regression framework by difference of convex functions programming. Appl. Intell. 45(2), 241–254 (2016)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  Google Scholar 

  324. Yin, P., Xin, J., Qi, Y.: Linear feature transform and enhancement of classification on deep neural network. (2016, Submitted)

  325. Ying, Y., Huang, K., Campbell, C.: Enhanced protein fold recognition through a novel data integration approach. BMC Bioinform. 10(1), 1–18 (2009)

    Article  Google Scholar 

  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)

  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)

  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)

  329. Yuille, A.L., Rangarajan, A.: The concave–convex procedure. Neural Comput. 15(4), 915–936 (2003)

    Article  MATH  Google Scholar 

  330. Zhang, K., Tsang, I.W., Kwok, J.T.: Maximum margin clustering made practical. IEEE Trans. Neural Netw. 20(4), 583–596 (2009)

    Article  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  336. Zhong, P.: Training robust support vector regression with smooth non-convex loss function. Optim. Methods Softw. 27(6), 1039–1058 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  342. Zou, H.: The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 2006(476), 1418–1429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  343. Zou, H., Li, R.: One-step sparse estimates in nonconcave penalized likelihood models. Ann. Stat. 36(4), 1509–1533 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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  1. Laboratory of Theoretical and Applied Computer Science (LITA), University of Lorraine, 3, rue Augustin Fresnel, 57 073, Metz Technopole, France

    Hoai An Le Thi

  2. Laboratory of Mathematics, National Institute for Applied Sciences - Rouen, 76801, Saint-Étienne-du-Rouvray Cedex, France

    Tao Pham Dinh

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Le Thi, H.A., Pham Dinh, T. DC programming and DCA: thirty years of developments. Math. Program. 169, 5–68 (2018). https://doi.org/10.1007/s10107-018-1235-y

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