Minimizing Compositions of Differences-of-Convex Functions with Smooth Mappings

References

• [1] Artina M, Fornasier M, Solombrino F (2013) Linearly constrained nonsmooth and nonconvex minimization. SIAM J. Optim. 23(3):1904–1937.Crossref, Google Scholar
• [2] Axelsson O, Lu H, Polman B (1994) On the numerical radius of matrices and its application to iterative solution methods. Linear Multilinear Algebra 37(1–3):225–238.Crossref, Google Scholar
• [3] Bauschke HH, Güler O, Lewis AS, Sendov HS (2001) Hyperbolic polynomials and convex analysis. Can. J. Math. 53(3):470–488.Crossref, Google Scholar
• [4] Bolte J, Sabach S, Teboulle M (2018) Nonconvex lagrangian-based optimization: Monitoring schemes and global convergence. Math. Oper. Res. 43(3):1210–1232.Link, Google Scholar
• [5] Boyd S, Balakrishnan V (1990) A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L∞-norm. Systems Control Lett. 15(1):1–7.Crossref, Google Scholar
• [6] Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, New York).Crossref, Google Scholar
• [7] Burger M, Osher S (2013) A Guide to the TV Zoo (Springer International Publishing, Cham, Switzerland), 1–70.Crossref, Google Scholar
• [8] Burke J (1985) Descent methods for composite nondifferentiable optimization problems. Math. Program, Serie A. 33:260–279.Crossref, Google Scholar
• [9] Byrd R, Nocedal J, Waltz R (2008) Steering exact penalty methods for nonlinear programming. Optim. Methods Softw. 23(2):197–213.Crossref, Google Scholar
• [10] Byrd R, Gould N, Nocedal J, Waltz R (2005) On the convergence of successive linear-quadratic programming algorithms. SIAM J. Optim. 16(2):471–489.Crossref, Google Scholar
• [11] Combari C, Elhilali Alaoui A, Levy A, Poliquin R, Thibault L (1998) Convex composite functions in banach spaces and the primal lower-nice property. Proc. Amer. Math. Soc. 126(12):3701–3708.Crossref, Google Scholar
• [12] Conn A, Pietrzykowski T (1977) A penalty function method converging directly to a constrained optimum. SIAM J. Numer. Anal. 14(2):348–375.Crossref, Google Scholar
• [13] Conn AR, Gould N, Toint P (2000) Trust Region Methods (SIAM, Philadelphia).Crossref, Google Scholar
• [14] Fan J, Li R (2004) Variable selection via nonconvex penalized likehood and its oracle properties. Ann. Statist. 32:407–499.Google Scholar
• [15] Fierro R, Golub G, Hansen P, O’Leary D (1997) Regularization by truncated total least squares. SIAM J. Sci. Comput. 18(1):1223–1241.Crossref, Google Scholar
• [16] Fletcher R (1987) Practical Methods of Optimization (John Wiley & Sons, New York).Google Scholar
• [17] Fletcher R, Sainz de la Maza E (1989) Nonlinear programming and nonsmooth optimization by successive linear programming. Math. Programming 43:235–256.Crossref, Google Scholar
• [18] Fukushima M, Mine H (1981) A generalized proximal point algorithm for certain nonconvex minimization problems. Internat. J. Systems Sci. 12:989–1000.Crossref, Google Scholar
• [19] Geiping J, Moeller M (2018) Composite optimization by nonconvex majorization-minimization. SIAM J. Imaging Sci. 11(4):2494–2528.Crossref, Google Scholar
• [20] Golub G, Hansen H, O’Leary D (1999) Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21(1):185–194.Crossref, Google Scholar
• [21] Han S, Mangasarian O (1979) Exact penalty functions in nonlinear programming. Math. Programming 17(3):251–269.Crossref, Google Scholar
• [22] He C, Watson GA (1997) An algorithm for computing the numerical radius. IMA J. Numer. Anal. 17(3):329–342.Crossref, Google Scholar
• [23] Hiriart-Urruty JB, Ye D (1995) Sensitivity analysis of all eigenvalues of a symmetric matrix. Numerische Maththematick 70(1):45–72.Crossref, Google Scholar
• [24] Ho VT, Le Thi HA, Pham Dinh T (2021) DCA-based algorithms for DC fitting. J. Comput. Appl. Math. 389:113353.Crossref, Google Scholar
• [25] Horn RA, Johnson CR (1991) Topics in Matrix Analysis (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [26] Huffel SV, Zha H (1991) The restricted total least squares problem: Formulation, algorithm, and properties. SIAM J. Matrix Anal. Appl. 12(2):292–309.Crossref, Google Scholar
• [27] Huynh VN, Théra M (2009) Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Programming 116:397–427.Crossref, Google Scholar
• [28] Jokar S, Pfetsch M (2008) Exact and approximate sparse solutions of underdetermined linear equations. SIAM J. Sci. Comput. 31:23–44.Crossref, Google Scholar
• [29] Jourani A (1998) The role of locally compact cones in nonsmooth analysis. Comm. Appl. Nonlinear Anal. 5:1–35.Google Scholar
• [30] Le Thi HA, Ho VT (2020) Online learning based on online DCA and application to online classification. Neural Comput. 32(4):759–793.Crossref, Google Scholar
• [31] Le Thi HA, Pham Dinh T (2003) Large scale molecular optimization from distance matrices by a D.C. optimization approach. SIAM J. Optim. 14(1):77–114.Crossref, Google Scholar
• [32] Le Thi HA, Pham Dinh T (2005) The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133:23–46.Crossref, Google Scholar
• [33] Le Thi HA, Pham Dinh T (2018) DC programming and DCA: Thirty years of developments. Math. Programming, Special Issue dedicated to 30-th birthday of DC programming and DCA: DC Programming – Theory, Algorithms and Applications 169(1):5–68.Google Scholar
• [34] Le Thi HA, Huynh VN, Pham Dinh T (2018) Convergence analysis of dc algorithm for dc programming with subanalytic data. J. Optim. Theory Appl. 179:103–126.Crossref, Google Scholar
• [35] Le Thi HA, Le HM, Phan DN, Tran B (2017) Stochastic DCA for the large-sum of non-convex functions problem and its application to group variable selection in classification. Proc. 34th Internat. Conf. Machine Learn., vol. 70 (JMLR), 3394–3403 ( JMLR.org).Google Scholar
• [36] Le Thi HA, Le HM, Phan DN, Tran B (2021) Novel DCA based algorithms for a special class of nonconvex problems with application in machine learning. Appl. Math. Comput. 409:125904.Crossref, Google Scholar
• [37] Le Thi HA, Pham Dinh T, Le HM, Vo XT (2015) DC approximation approaches for sparse optimization. Eur. J. Oper. Res. 244:26–44.Crossref, Google Scholar
• [38] Lewis AS, Wright SJ (2016) A proximal method for composite minimization. Math. Programming 158(1):501–546.Crossref, Google Scholar
• [39] Liu T, Pong TK, Takeda A (2019) A successive difference-of-convex approximation method for a class of nonconvex nonsmooth optimization problems. Math. Programming 176(1):339–367.Crossref, Google Scholar
• [40] Mangasarian OL (1999) Minimum-support solutions of polyhedral concave programs. Optimization 45:149–162.Crossref, Google Scholar
• [41] Markovsky I, Huffel SV (2007) Overview of total least-squares methods. Signal Processing 87:2283–2302.Crossref, Google Scholar
• [42] Mengi E, Overton ML (2005) Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix. IMA J. Numer. Anal. 25(4):648–669.Crossref, Google Scholar
• [43] Mordukhovich BS (2006) Variational analysis and generalized differentiation. I. Basic theory, vol. 330. Grundlehren der Mathematischen Wissenschaften (Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg).Google Scholar
• [44] Pham Dinh T, Le Thi HA (1997) Convex analysis approach to DC programming: Theory, algorithms and applications. Acta Math. Vietnam. 22(1):289–355.Google Scholar
• [45] Pham Dinh T, Le Thi HA (1998) A D.C. optimization algorithms for solving the trust region subproblem. SIAM J. Optim. 8(2):476–505.Crossref, Google Scholar
• [46] Pham Dinh T, Le Thi HA (2014) Recent advances in DC programming and DCA. Nguyen NT, Le-Thi H, eds., Transactions on Computational Intelligence XIII (Springer Berlin Heidelberg, Berlin, Heidelberg), 1–37.Crossref, Google Scholar
• [47] Pham Dinh T, Huynh VN, Le Thi HA, Ho VT (2022) Alternating DC algorithm for partial DC programming problems. J. Global Optim. 82:897–928.Crossref, Google Scholar
• [48] Pillo G, Facchinei F (1989) Exact penalty functions for nondifferentiable programming problems. Clarke F, Dem’yanov V, Giannessi F, eds., Nonsmooth Optimization and Related Topics (Springer, Berlin), 89–105.Crossref, Google Scholar
• [49] Poliquin RA (1991) Integration of subdifferentials of nonconvex functions. Nonlinear Anal. Th. Meth. Appl. 17:385–398.Crossref, Google Scholar
• [50] Poliquin RA, Rockafellar RT, Thibault L (2000) Local differentiability of distance functions. Trans. Amer. Math. Soc. 352:5231–5249.Crossref, Google Scholar
• [51] Rockafellar RT (1970) Convex Analysis (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• [52] Rockafellar RT, Wets RJB (1998) Variational Analysis (Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg).Crossref, Google Scholar
• [53] Sabach S, Teboulle M (2019) Lagrangian methods for composite optimization. Kimmel R, Tai XC, eds., Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, vol. 20, Handbook of Numerical Analysis (Elsevier, Amsterdam), 401–436.Crossref, Google Scholar
• [54] Sagastizábal C (2013) Composite proximal bundle method. Math. Program. Serie B. 140:189–233.Crossref, Google Scholar
• [55] Solodov MV (2004) On the sequential quadratically constrained quadratic programming methods. Math. Oper. Res. 29:64–79.Link, Google Scholar
• [56] Sra S, Nowozin S, Wright SJ (2011) Optimization for Machine Learning (MIT Press, Cambridge, MA).Crossref, Google Scholar
• [57] Uhlig F (2009) Geometric computation of the numerical radius of a matrix. Numer. Algorithms 52:335–353.Crossref, Google Scholar
• [58] Watson GA (1996) Computing the numerical radius. Linear Algebra Appl. 234:163–172.Crossref, Google Scholar
• [59] Wright S, Nowak R, Figueired M (2009) Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57:2479–2493.Crossref, Google Scholar
• [60] Wright SJ (1990) Convergence of an inexact algorithm for composite nonsmooth optimization. IMA J. Numer. Anal. 9:299–321.Crossref, Google Scholar
• [61] Xu Y, Qi Q, Lin Q, Jin R, Yang T (2019) Stochastic optimization for DC functions and non-smooth non-convex regularizers with non-asymptotic convergence. Chaudhuri K, Salakhutdinov R, eds. Proc. 36th Internat. Conf. Machine Learn., vol. 97 (PMLR, New York), 6942–6951.Google Scholar
• [62] Zangwill WI (1967) Nonlinear programming via penalty functions. Management Sci. 13:344–358.Link, Google Scholar
• [63] Zhang CH (2010) Nearly unbiased variable selectionunder minimax concave penalty. Ann. Statist. 38:894–942.Crossref, Google Scholar