[1] Aragón Artacho F and Borwein J. Global convergence of a non-convex Douglas-Rachford iteration[J]. Journal of Global Optimization., 2013, 57(3):753-769.[2] Attouch H, Briceño-Arias L M and Combettes P L. A parallel splitting method for coupled monotone inclusions[J]. SIAM Journal on Control optimization., 2010, 48(5):3246-3270.[3] Attouch H, Briceño-Arias L M and Combettes P L. A strongly convergent primal-dual method for nonoverlapping domain decomposition[J]. Numerische Mathematik., 2016, 133(3):443-470.[4] Attouch H, Bolte J and Svaiter B F. Convergence of descent methods for semi-algebraic and tame problems:proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[J]. Mathematical Programming., 2013, 137:91-129.[5] Attouch H, Peypouquet J and Redont P. Backward-forward algorithms for structured monotone inclusions in Hilbet spaces[J]. Journal of Mathematical Analysis and Applications., 2018, 457:1095-1117.[6] Attouch H, Redont P and Soubeyran A. A new class of alternating proximal minimization algorithms with costs-to-move[J]. SIAM Journal on Optimization., 2007, 18:1061-1081.[7] Attouch H and Thera M. A general duality principle for the sum of two operators[J]. Journal of Convex Analysis., 1996, 3:1-24.[8] Baillon J B and Haddad G. Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones[J]. Israel Journal of Mathematics., 1977, 26:137-150.[9] Bauschke H and Borwein J. Dykstra's alternating projection algorithm for two sets[J]. Journal of Approximation Theory., 1994, 79(3):418-443.[10] Bauschke H and Browein J. On projection algorithms for solving convex feasibility problems[J]. SIAM Review., 1996, 38(3):367-426.[11] Bauschke H and Borwein J. On the convergence of von Neumann's alternating projection algorithm for two sets[J]. Set-Valued Analysis., 1993, 1(2):185-212.[12] Bauschke H H and Combettes P. Convex analysis and Monotone operator Theory in Hilbert spaces[M]. (2011).[13] Bauschke H, Combettes P and Luke D. Hybrid projection-reflection method for phase retrieval[J]. Journal of the Optical Society of America A., 2003, 20(6):1025-1034.[14] Bauschke H, Combettes P and Luke D. Phase retrieval, error reduction algorithm, and Fienup variants:a view from convex optimization[J]. Journal of the Optical Society of America A., 2002, 19(7):1334-1345.[15] Beck A and Teboulle M. A fast iterative shrinkage-thresholding algorithm for linear inverse problem[J]. SIAM Journal on Imaging Sciences., 2009, 2(1):183-202.[16] Bot R I and Hendrich C. A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators[J]. SIAM Journal on Optimization., 2013, 23(4):2541-2565.[17] Boyd S, Parikh N, Chu E, Peleato B and Eckstein J. Distributed optimization and statistical learning via the alternating direction method of multipliers[J]. Foundations and Trends in Machine Learning., 2011, 3:1-122.[18] Bregman L. The method of successive projection for finding a common point of convex sets. Dokl. Akad. Nauk SSSR., 1965, 6:688-692.[19] Browder F E. Nonlinear elliptic boundary value problems[J]. Bulletin of the American Mathematical Society., 1963, 69(6):862-874.[20] Browder F E. The solvability of non-linear function equations[J]. Duke Mathematical Journal., 1963, 30(4):557-566.[21] Browder F E. Variational boundary value problems for quasi-linear ellptic equations of arbitrary order[J]. Proceedings of the National Academy of Sciences of the United States of America., 1963, 50(1):31-37.[22] Browein J and Sims B. The Douglas-Rachford algorithm in absence of convexity[M]. In:FixedPoint Algorithms for Inverse Problems in Science and Engineering., 2011, 93-109.[23] Chambolle A and Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging[J]. Journal of Mathematical Imaging and Vision., 2011, 40:120-145.[24] Chen L, Sun D F and Toh K C. A note on the convergence of ADMM for linearly constrained convex optimization problems[J]. Computational Optimization and Applications., 2017, 66(2):327-343.[25] Combettes P L. Solving monotone inclusions via compositions of nonexpansive averaged operators[J]. Optimization., 2004, 53:475-504.[26] Combettes P L and Briceño-Arias L M. A monotone + skew splitting model for composite monotone inclusion in duality[J]. SIAM Journal on Optimization., 2010, 21(4):1230-1250.[27] Combettes P L and Pesquet J C. Proximal splitting methods in signal processing[M]. In:FixedPoint Algorithms for Inverse problems in Science and Engineering., 2011, 185-212.[28] Combettes P L and Vu B C. Variable metric forward-backward splitting with applications to monotone inclusions in duality[J]. Optimization., 2014, 63:1289-1318.[29] Combettes P L and Vu B C. Variable metric quasi-Fejer monotonicity[J]. Nonlinear Analysis., 2013, 78:17-31.[30] Combettes P L and Wajs V R. Signal recovery by proximal forward-backward splitting[J]. SIAM Journal on Multiscale Modeling and Simulation., 2005, 4:1168-1200.[31] Corman E and Yuan X M. A generalized proximal point algorithm and its convergence rate. SIAM Journal on Optimization., 2014, 24(4):1614-1638.[32] Douglas J and Rachford H H. On the numerical solution of heat conduction problems in two or three space variables[J]. Transactions of the Ameican Mathematical Society., 1956, 82:421-439.[33] Dykstra R. An algorithm for restricted least squares regression[J]. Publications of the American Statistical Association., 1983, 78(384):837-842.[34] Eckstein J. Splitting methods for monotone operators with applications to parallel optimization[M]. Ph.D. thesis, MIT, 1989.[35] Eckstein J and Bertsekas D P. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators[J]. Mathematical Programming., 1992, 55:293-318.[36] Eckstein J and Wang Yao. Understanding the convergence of the alternating direction method of multipliers:theoretical and computational perspectives[J]. Pacific Journal of Optimization., 2015, 11(4):619-644.[37] Elser V, Rankenburg I and Thibault P. Searching with iterated maps[J]. Proceedings of the National Academy of Sciences of the United States of America., 2007, 104(2):418-423.[38] Esser E, Zhang X Q and Chan T. A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[J]. SIAM Journal on Imaging Sciences., 2010, 3(4):1015-1046.[39] Facchinei F and Pang J S. Finite-Dimensional Variational Inequalities and Complementarity Problems[M]. Springer, Berlin (2003).[40] Fazel M, Pong T K, Sun D F and Tseng P. Hankel matrix rank minimization with applications to system identification and realization[J]. SIAM Journal on Matrix Analalysis and Applications., 2013, 34(3):946-977.[41] Fortin M and Glowinski. On decomposition-coordination methods using an augmented Lagrangian[M]. In Augmented Lagragian Methods:Applications to the Solutions of Boundary-Value Problems., M. Fortin and R. Glowinski (eds.), North-Holland, Amsterdam, 1983, 97-164.[42] Gabay D. Applications of the method of multipliers to variational inequalities[M]. In Augmented Lagrangian methods:Applications to the Numerical Solution of Boundary-Value Problems., Fortin M, Glowinski R. (eds), North-Holland, Amsterdam (1983), 299-331.[43] Glowinski R. On alternating direction methods of multipliers:a historical perspective[M]. In:Fitzgibbon, W., Kuznetsov, Y.A., Neittaanmaki, P., Pironneau, O. (eds.) Modeling, Simulation and Optimization for Science and Technology, 59-82. Springer, Netherlands (2014).[44] Glowinski R and Marroco A. Sur l'approximation, par elements finis d'ordre un, et la resolution, par penalisation-dualité, d'une classe de problems de Dirichlet non lineares[J]. Rev. Française Informat. Recherche Opérationnelle., 1975, 9:41-76.[45] Goldburg M and Marks Ⅱ R J. Signal synthesis in the presence of an inconsistent set of constraints[J]. IEEE Transactions on Circuits and Systems., 1985, 32:647-663.[46] Goldstein A A. Convex programming in Hilbert space[J]. Bulletin of the American Mathematical Society., 1964, 70:709-710.[47] Golshtein E G and Tretyakov N V. Modified Lagrangians and Monotone Maps in Optimization[M]. John Wiley, New York (1996).[Translation of Modified Lagrangian Functions:Theory and Related Optimization Techniques, Nauka, Moscow (1989).][48] Gravel S and Elser V. Divide and concur:a general approach to constraint satisfaction[J]. Phys Rev E Stat Nonlin Soft Matter Phys., 2008, 78(3):036706.[49] Guo K, Han D R, David Wang Z W and Wu T T. Convergence of ADMM for multi-block nonconvex separable optimization models[J]. Frontier of Mathematics in China., 2017, 12(5):1139-1162.[50] Guo K, Han D R and Wu T T. Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints[J]. International Journal of Computer Mathematics., 2017, 94:1653-1669.[51] Guo K, Han D R and Wu T T. Convergence analysis for optimization problems with nonseparable nonconvex objective and linear constraints[J]. Pacific Journal on Optimization., Accepted.[52] Guo K, Han D R and Yuan X M. Convergenece analysis of Douglas-Rachfod splitting method for "strongly+weakly" convex programming[J]. SIAM Journal on Numerical Analysis., 2017, 55(4):1549-1577.[53] Guo K and Han D R. A note on the Douglas-Rachford splitting method for optimization problems involving weakly convex functions[J]. Journal of Global optimization. https://doi.org/10.1007/s10898-018-0660-z.[54] Halperin I. The product of projection operators[J]. Acta Scientiarum Mathematicarum., 1962, 23:96-99.[55] Han D R, He H J, Yang H and Yuan X M. A customized Douglas-Rachford splitting algorithm for separable convex minimization with linear constraints[J]. Numerische Mathematik., 2014, 127(1):167-200.[56] Hartman P and Stampacchia G. On some non-linear elliptic differential-functional equations[J]. Acta Mathematica., 1966, 115:271-310.[57] He B S, Liu H, Wang Z R and Yuan X M. A strictly contractive Peaceman-Rachford splitting method for convex programming[J]. SIAM Journal on Optimization., 2014, 24(3):1101-1140.[58] Kachurovskii R I. Monotone operators and convex functionals[J]. Uspekhi Mat Nauk., 1960, 15(4):213-215.[59] Kinderlehrer D and Stampacchia G. An Introduction to Variationa Inequalities and their Applications[M]. Academic Press, New York (1980).[60] Konnov I V. Combined relaxation methods for finding equilibrium point and solving related problems[J]. Russian Mathematics., 1993, 37:46-53.[61] Konnov I V. Equilibrium Models and Variational Inequalities[M]. Elsevier, Amsterdam (2007).[62] Korpelevich G M. The extragradient method for finding saddle points and other problems[J]. Ekonomie i Mathematik Metody., 1976, 12:747-756.[63] Krasnosel'skii M A. Two remarks on the method of successive approximations[J]. Uspekhi Mat Nauk., 1955, 10:123-127.[64] Lemaire B. The proximal algorithm[M]. In:Penot J P, (Ed.), New Methods in Optimization and their Industrial Uses, International Series of Numerical Mathematics, 87, 73-87. Birkhauser, Boston, MA.[65] Levitin E S and Polyak B T. Constrained minimization methods[J]. USSR Computational Mathematics and Mathematical Physics., 1966, 6:1-50.[66] Li G Y and Pong T K. Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems[J]. Mathematical Programming., 2016, 159:371-401.[67] Li G Y and Pong T K. Global convergence of splitting methods for nonconvex composite optimization[J]. SIAM Journal on Optimization., 2015, 25:2434-2460.[68] Lions P L and Mercier B. Splitting algorithms for the sum of two nonlinear operators[J]. SIAM Journal on Numerical Analysis., 1979, 16:964-979.[69] Martinet B. Determination approchee d'un point fixe d'une application pseudo-constractante[J]. C. R. Acad. Sci. Paris Ser. A., 1972, 274:163-165.[70] Martinet B. Regularisation d'inequations variationnelles par approximations successives[J]. Rev.francaise Informat.recherche Operationnelle., 1970, 4(3):154-158.[71] Minty G J. Monotone networks[J]. Proceedings of the Royal Society A., 1960, 257(1289):194-212.[72] Minty G J. Monotone (nonlinear) operators in Hilbert space[J]. Duke Mathematical Journal., 1962, 29:341-346.[73] Minty G J. On the maximal domain of a "monotone" function[J]. Michigan Mathematical Journal., 1961, 8(2):135-137.[74] Minty G J. On the monotonicity of the gradient of a convex functin[J]. Pacific Journal of Mathematics., 1964, 14(1), 243-247.[75] Nesterov Y E. A method for solving convex programming problem with convergence rate O(1/k2)[J]. Dokl. Akad. Nauk SSSR., 1983, 269:543-547.[76] Nguyen Q V. Variable quasi-Bregman monotone sequence[J]. Numerical Algorithms., 2016, 73:1107-1130.[77] O'Connor D and Vandenberghe L. On the equivalence of the primal-dual hybrid gradient method and Douglas-Rachford splitting[J]. Mathematical Programming., (2018). https://doi.org/10.1007/s10107-018-1321-1.[78] Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings[J]. Bulletin of the American Mathematical Society., 1967, 73:591-597.[79] Passty G B. Ergodic convergence to a zero of the sum of monotone operators in Hilbert space[J]. Journal of Mathematical Analysis and Applications., 1979, 72:383-390.[80] Peaceman D W and Rachford H H. The numerical solution of parabolic and elliptic differential equations[J]. Journal of the Society for Industrial and Applied Mathematics., 1955, 3:28-41.[81] Pennanen T. A splitting method for composite mappings[J]. Numerical Functional Analysis and Optimization, 23:875-890.[82] Rockafellar R T. Characterization of the subdifferentials of convex functions[J]. Pacific Journal of Mathematics., 1966, 17(3):497-510.[83] Rockafellar R T. Lagrange multipliers and optimality[J]. SIAM Review., (1993), 35(2):183-238.[84] Rockafellar R T. Monotone operators and the proximal point algorithm[J]. SIAM Journal on Control optimization., 1976, 14(5):877-898.[85] Rockafellar R T and Wets R J-B. Variational Analysis[M]. Springer-Verlag, Berlin (1998).[86] Sibony M. Méthodes itératives pour les équations et inéquations aux dérivés partielles nonlinéares de type monotone[J]. Calcolo., 1970, 7:65-183.[87] Spingarn J E. Applications of the method of parital inverses to convex programming:decomposition[J]. Mathematical Programming., 1985, 32:199-223.[88] Tseng P. A modified forward-backward splitting method for maximal monotone mappings[J]. SIAM Journal on Control optimization., 2000, 38:431-446.[89] Tseng P. Applications of a splitting algorithm to decomposition in convex programming and vaiational inequalities[J]. SIAM Journal on Control and optimization., 1991, 29:119-138.[90] Tseng P. Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming[J]. Mathematical Programming., 1990, 48:249-263.[91] Tian W Y and Yuan X M. Faster alternating direction method of multipliers with O(1/n2) convergence rates[J]. Mathematics of Computation., In revision.[92] Varga R S. Matrix Iterative Analysis[M]. 2nd Edn. Springer-Verlas, New York.[93] Von Neumann J. Functional Operators, vol. Ⅱ. The Geometry of Orthogonal Spaces[M]. Princeton University Press, Princeton (1950).[94] Wang F H, Xu Z B and Xu H K. Convergence of alternating direction method with multipliers for non-convex composite problems[J]. arXiv preprint arXiv:1410.8625., (2014). |