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The geometry of monotone operator splitting methods

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  04 September 2024

Patrick L. Combettes Open the ORCID record for Patrick L. Combettes [Opens in a new window]
Patrick L. Combettes*
Affiliation:
North Carolina State University, Department of Mathematics, Raleigh, NC 27695-8205, USA E-mail: plc@math.ncsu.edu
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Abstract

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We propose a geometric framework to describe and analyse a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones.

MSC classification

Primary: 47H05: Monotone operators and generalizations 90C48: Programming in abstract spaces 90C46: Optimality conditions, duality
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 90C25: Convex programming
Type
Research Article
Information
Acta Numerica , Volume 33 , July 2024 , pp. 487 - 632
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

*

This work was supported by the National Science Foundation under grant CCF-2211123.

References

Acker, F. and Prestel, M. A. (1980), Convergence d’un schéma de minimisation alternée, Ann. .Fac. Sci. Toulouse V. Sér. Math 2, 1–9.10.5802/afst.541CrossRefGoogle Scholar
Adly, S., Hantoute, A. and Le, B. K. (2017), Maximal monotonicity and cyclic monotonicity arising in nonsmooth Lur’e dynamical systems, J. Math. Anal. Appl. 448, 691–706.10.1016/j.jmaa.2016.11.025CrossRefGoogle Scholar
Agmon, S. (1954), The relaxation method for linear inequalities, Canad. J. Math. 6, 382–392.10.4153/CJM-1954-037-2CrossRefGoogle Scholar
Alber, Y. and Ryazantseva, I. (2006), Nonlinear Ill-Posed Problems of Monotone Type, Springer.Google Scholar
Alduncin, G. (2005), Composition duality principles for mixed variational inequalities, Math. Comput. Modelling 41, 639–654.10.1016/j.mcm.2004.10.022CrossRefGoogle Scholar
Alduncin, G. (2023), Multidomain optimal control of variational subpotential mixed evolution inclusions, Appl. Math. Optim. 88, art. 35.10.1007/s00245-023-10010-8CrossRefGoogle Scholar
Alghamdi, M. A., Alotaibi, A., Combettes, P. L. and Shahzad, N. (2014), A primal–dual method of partial inverses for composite inclusions, Optim. Lett. 8, 2271–2284.10.1007/s11590-014-0734-xCrossRefGoogle Scholar
Alimohammady, M., Ramazannejad, M. and Roohi, M. (2014), Notes on the difference of two monotone operators, Optim. Lett. 8, 81–84.10.1007/s11590-012-0537-xCrossRefGoogle Scholar
Alotaibi, A., Combettes, P. L. and Shahzad, N. (2014), Solving coupled composite monotone inclusions by successive Fejér approximations of their Kuhn–Tucker set, SIAM J. Optim. 24, 2076–2095.10.1137/130950616CrossRefGoogle Scholar
Alotaibi, A., Combettes, P. L. and Shahzad, N. (2015), Best approximation from the Kuhn–Tucker set of composite monotone inclusions, Numer. .Funct. Anal. Optim 36, 1513–1532.10.1080/01630563.2015.1077864CrossRefGoogle Scholar
Anderson, W. N. Jr and Trapp, G. E. (1976), A class of monotone operator functions related to electrical network theory, Linear Algebra Appl. 15, 53–67.10.1016/0024-3795(76)90078-1CrossRefGoogle Scholar
Antipin, A. S. (1976), On a method for convex programs using a symmetrical modification of the Lagrange function, Èkonom. i Mat. Metody 12, 1164–1173.Google Scholar
Aoyama, K., Kimura, Y. and Takahashi, W. (2008), Maximal monotone operators and maximal monotone functions for equilibrium problems, J. Convex Anal. 15, 395–409.Google Scholar
Aragón-Artacho, F. J., Boţ, R. I. and Torregrosa-Belén, D. (2023), A primal–dual splitting algorithm for composite monotone inclusions with minimal lifting, Numer. Algorithms 93, 103–130.10.1007/s11075-022-01405-9CrossRefGoogle ScholarPubMed
Argyriou, A., Foygel, R. and Srebro, N. (2012), Sparse prediction with the k-support norm, in Advances in Neural Information Processing Systems 25 (Pereira, F. et al., eds), Curran Associates, pp. 1457–1465.Google Scholar
Attouch, H. and Cabot, A. (2020), Convergence of a relaxed inertial proximal algorithm for maximally monotone operators, Math. Program. A184, 243–287.10.1007/s10107-019-01412-0CrossRefGoogle Scholar
Attouch, H. and Théra, M. (1996), A general duality principle for the sum of two operators, J. Convex Anal. 3, 1–24.Google Scholar
Attouch, H., Bolte, J., Redont, P. and Soubeyran, A. (2008), Alternating proximal algorithms for weakly coupled convex minimization problems: Applications to dynamical games and PDE’s, J. Convex Anal. 15, 485–506.Google Scholar
Attouch, H., Briceño-Arias, L. M. and Combettes, P. L. (2010), A parallel splitting method for coupled monotone inclusions, SIAM J. Control Optim. 48, 3246–3270.10.1137/090754297CrossRefGoogle Scholar
Attouch, H., Briceño-Arias, L. M. and Combettes, P. L. (2016), A strongly convergent primal–dual method for nonoverlapping domain decomposition, Numer. Math. 133, 433–470.10.1007/s00211-015-0751-4CrossRefGoogle Scholar
Attouch, H., Buttazzo, G. and Michaille, G. (2014), Variational Analysis in Sobolev and BV Spaces, second edition, SIAM.10.1137/1.9781611973488CrossRefGoogle Scholar
Attouch, H., Cabot, A., Frankel, P. and Peypouquet, J. (2011), Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE’s, Nonlinear Anal. 74, 7455–7473.10.1016/j.na.2011.07.066CrossRefGoogle Scholar
Attouch, H., Peypouquet, J. and Redont, P. (2018), Backward–forward algorithms for structured monotone inclusions in Hilbert spaces, J. Math. Anal. Appl. 457, 1095–1117.10.1016/j.jmaa.2016.06.025CrossRefGoogle Scholar
Aubin, J.-P. and Cellina, A. (1984), Differential Inclusions: Set-Valued Maps and Viability Theory, Springer.10.1007/978-3-642-69512-4CrossRefGoogle Scholar
Aujol, J.-F. and Chambolle, A. (2005), Dual norms and image decomposition models, Int. J. Comput. Vision 63, 85–104.10.1007/s11263-005-4948-3CrossRefGoogle Scholar
Baillon, J.-B. and Haddad, G. (1977), Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones, Israel J. Math. 26, 137–150.10.1007/BF03007664CrossRefGoogle Scholar
Bakušinskiĭ, A. B. and Polyak, B. T. (1974), The solution of variational inequalities, Soviet Math. Dokl. 15, 1705–1710.Google Scholar
Banert, S., Boţ, R. I. and Csetnek, E. R. (2021), Fixing and extending some recent results on the ADMM algorithm, Numer. Algorithms 86, 1303–1325.10.1007/s11075-020-00934-5CrossRefGoogle ScholarPubMed
Barbu, V. (2010), Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer.10.1007/978-1-4419-5542-5CrossRefGoogle Scholar
Bartz, S., Bauschke, H. H., Moffat, S. M. and Wang, X. (2016), The resolvent average of monotone operators: Dominant and recessive properties, SIAM J. Optim. 26, 602–634.10.1137/15M1020964CrossRefGoogle Scholar
Bauschke, H. H. and Borwein, J. M. (1994), Dykstra’s alternating projection algorithm for two sets, J. Approx. Theory 79, 418–443.10.1006/jath.1994.1136CrossRefGoogle Scholar
Bauschke, H. H. and Combettes, P. L. (2001), A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res. 26, 248–264.10.1287/moor.26.2.248.10558CrossRefGoogle Scholar
Bauschke, H. H. and Combettes, P. L. (2008), A Dykstra-like algorithm for two monotone operators, Pacific J. Optim. 4, 383–391.Google Scholar
Bauschke, H. H. and Combettes, P. L. (2017), Convex Analysis and Monotone Operator Theory in Hilbert Spaces, second edition, Springer.10.1007/978-3-319-48311-5CrossRefGoogle Scholar
Bauschke, H. H. and Moursi, W. M. (2017), On the Douglas–Rachford algorithm, Math. Program. A164, 263–284.10.1007/s10107-016-1086-3CrossRefGoogle Scholar
Bauschke, H. H., Bolte, J. and Teboulle, M. (2017), A descent lemma beyond Lipschitz gradient continuity: First-order methods revisited and applications, Math. Oper. Res. 42, 330–348.10.1287/moor.2016.0817CrossRefGoogle Scholar
Bauschke, H. H., Borwein, J. M. and Combettes, P. L. (2003), Bregman monotone optimization algorithms, SIAM J. Control Optim. 42, 596–636.10.1137/S0363012902407120CrossRefGoogle Scholar
Bauschke, H. H., Bùi, M. N. and Wang, X. (2019), On sums and convex combinations of projectors onto convex sets, J. Approx. Theory 242, 31–57.10.1016/j.jat.2019.02.001CrossRefGoogle Scholar
Bauschke, H. H., Combettes, P. L. and Reich, S. (2005), The asymptotic behavior of the composition of two resolvents, Nonlinear Anal. 60, 283–301.10.1016/j.na.2004.07.054CrossRefGoogle Scholar
Bauschke, H. H., Deutsch, F. and Hundal, H. (2009), Characterizing arbitrarily slow convergence in the method of alternating projections, Int. Trans. Oper. Res. 16, 413–425.10.1111/j.1475-3995.2008.00682.xCrossRefGoogle Scholar
Bauschke, H. H., Koch, V. R. and Phan, H. M. (2016), Stadium norm and Douglas–Rachford splitting: A new approach to road design optimization, Oper. Res. 64, 201–218.10.1287/opre.2015.1427CrossRefGoogle Scholar
Bauschke, H. H., Matoušková, E. and Reich, S. (2004), Projection and proximal point methods: Convergence results and counterexamples, Nonlinear Anal. 56, 715–738.10.1016/j.na.2003.10.010CrossRefGoogle Scholar
Beck, A. and Teboulle, M. (2009a), Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process. 18, 2419–2434.10.1109/TIP.2009.2028250CrossRefGoogle ScholarPubMed
Beck, A. and Teboulle, M. (2009b), A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2, 183–202.10.1137/080716542CrossRefGoogle Scholar
Beck, A. and Teboulle, M. (2010), Gradient-based algorithms with applications to signal recovery problems, in Convex Optimization in Signal Processing and Communications (Palomar, D. P. and Eldar, Y. C., eds), Cambridge University Press, pp. 42–88.Google Scholar
Becker, S. R. and Combettes, P. L. (2014), An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery, J. Nonlinear Convex Anal. 15, 137–159.Google Scholar
Bednarczuk, E. M., Jezierska, A. and Rutkowski, K. E. (2018), Proximal primal–dual best approximation algorithm with memory, Comput. Optim. Appl. 71, 767–794.10.1007/s10589-018-0031-1CrossRefGoogle Scholar
Belgioioso, G., Nedich, A. and Grammatico, S. (2021), Distributed generalized Nash equilibrium seeking in aggregative games on time-varying networks, IEEE Trans. Automat. Control 66, 2061–2075.10.1109/TAC.2020.3005922CrossRefGoogle Scholar
Bellman, R., Kalaba, R. E. and Lockett, J. A. (1966), Numerical Inversion of the Laplace Transform: Applications to Biology, Economics Engineering, and Physics, Elsevier.Google Scholar
Beltrami, E. (1972), A note regarding abstract operators and passive networks, Quart. Appl. Math. 30, 369–370.10.1090/qam/99721CrossRefGoogle Scholar
Benning, M. and Burger, M. (2018), Modern regularization methods for inverse problems, Acta Numer. 27, 1–111.10.1017/S0962492918000016CrossRefGoogle Scholar
Berge, C. and Ghouila-Houri, A. (1962), Programmes, Jeux, et Réseaux de Transport, Dunod. English translation: Programming, Games and Transportation Networks, Wiley (1965).Google Scholar
Bertero, M., Bindi, D., Boccacci, P., Cattaneo, M., Eva, C. and Lanza, V. (1997), Application of the projected Landweber method to the estimation of the source time function in seismology, Inverse Problems 13, 465–486.10.1088/0266-5611/13/2/017CrossRefGoogle Scholar
Bertsekas, D. P. (1998), Network Optimization: Continuous and Discrete Models, Athena Scientific.Google Scholar
Birkhoff, G. and Varga, R. S. (1959), Implicit alternating direction methods, Trans. Amer. Math. Soc. 92, 13–24.10.1090/S0002-9947-1959-0105814-4CrossRefGoogle Scholar
Blum, E. and Oettli, W. (1994), From optimization and variational inequalities to equilibrium problems, Math. Student 63, 123–145.Google Scholar
Börgens, E. and Kanzow, C. (2021), ADMM-type methods for generalized Nash equilibrium problems in Hilbert spaces, SIAM J. Optim. 31, 377–403.10.1137/19M1284336CrossRefGoogle Scholar
Borwein, J. M. (2010), Fifty years of maximal monotonicity, Optim. Lett. 4, 473–490.10.1007/s11590-010-0178-xCrossRefGoogle Scholar
Boţ, R. I. (2010), Conjugate Duality in Convex Optimization, Springer.10.1007/978-3-642-04900-2CrossRefGoogle Scholar
Boţ, R. I. and Csetnek, E. R. (2019), ADMM for monotone operators: Convergence analysis and rates, Adv. Comput. Math. 45, 327–359.10.1007/s10444-018-9619-3CrossRefGoogle Scholar
Boţ, R. I. and Hendrich, C. (2013), A Douglas–Rachford type primal–dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim. 23, 2541–2565.10.1137/120901106CrossRefGoogle Scholar
Boţ, R. I. and Hendrich, C. (2014), Convergence analysis for a primal–dual monotone+skew splitting algorithm with applications to total variation minimization, J. Math. Imaging Vision 49, 551–568.10.1007/s10851-013-0486-8CrossRefGoogle Scholar
Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2010), Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Machine Learn. 3, 1–122.10.1561/2200000016CrossRefGoogle Scholar
Boyle, J. P. and Dykstra, R. L. (1986), A method for finding projections onto the intersection of convex sets in Hilbert spaces, in Advances in Order Restricted Statistical Inference, Vol. 37 of Lecture Notes in Statistics, Springer, pp. 28–47.10.1007/978-1-4613-9940-7_3CrossRefGoogle Scholar
Bredies, K., Chenchene, E., Lorenz, D. A. and Naldi, E. (2022), Degenerate preconditioned proximal point algorithms, SIAM J. Optim. 32, 2376–2401.10.1137/21M1448112CrossRefGoogle Scholar
Brègman, L. M. (1965), The method of successive projection for finding a common point of convex sets, Soviet Math. Dokl. 6, 688–692.Google Scholar
Brègman, L. M. (1967), The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys. 7, 200–217.10.1016/0041-5553(67)90040-7CrossRefGoogle Scholar
Brézis, H. (1966), Les opérateurs monotones, Séminaire Choquet: Initiation à l’Analyse 5, 1–33.Google Scholar
Brézis, H. (1971), Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis (Zarantonello, E. H., ed.), Academic, pp. 101–156.10.1016/B978-0-12-775850-3.50009-1CrossRefGoogle Scholar
Brézis, H. (1973), Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier.Google Scholar
Brézis, H. and Browder, F. (1998), Partial differential equations in the 20th century, Adv. .Math 135, 76–144.10.1006/aima.1997.1713CrossRefGoogle Scholar
Brézis, H. and Lions, P. L. (1978), Produits infinis de résolvantes, Israel J. Math. 29, 329–345.10.1007/BF02761171CrossRefGoogle Scholar
Briceño-Arias, L. M. (2012), A Douglas–Rachford splitting method for solving equilibrium problems, Nonlinear Anal. 75, 6053–6059.10.1016/j.na.2012.06.014CrossRefGoogle Scholar
Briceño-Arias, L. M. and Combettes, P. L. (2009), Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery, Numer. Math. Theory Methods Appl. 2, 485–508.Google Scholar
Briceño-Arias, L. M. and Combettes, P. L. (2011), A monotone+skew splitting model for composite monotone inclusions in duality, SIAM J. Optim. 21, 1230–1250.10.1137/10081602XCrossRefGoogle Scholar
Briceño-Arias, L. M. and Combettes, P. L. (2013), Monotone operator methods for Nash equilibria in non-potential games, in Computational and Analytical Mathematics (Bailey, D. et al., eds), Springer, pp. 143–159.10.1007/978-1-4614-7621-4_9CrossRefGoogle Scholar
Briceño-Arias, L. M. and Davis, D. (2018), Forward–backward-half forward algorithm for solving monotone inclusions, SIAM J. Optim. 28, 2839–2871.10.1137/17M1120099CrossRefGoogle Scholar
Briceño-Arias, L. M. and Roldán, F. (2023), Primal–dual splittings as fixed point iterations in the range of linear operators, J. Global Optim. 85, 847–866.10.1007/s10898-022-01237-wCrossRefGoogle Scholar
Briceño-Arias, L. M., Chierchia, G., Chouzenoux, E. and Pesquet, J.-C. (2019), A random block-coordinate Douglas–Rachford splitting method with low computational complexity for binary logistic regression, Comput. Optim. Appl. 72, 707–726.10.1007/s10589-019-00060-6CrossRefGoogle Scholar
Briceño-Arias, L. M., Combettes, P. L., Pesquet, J.-C. and Pustelnik, N. (2011), Proximal algorithms for multicomponent image recovery problems, J. Math. Imaging Vision 41, 3–22.10.1007/s10851-010-0243-1CrossRefGoogle Scholar
Briceño-Arias, L. M., Deride, J., López-Rivera, S. and Silva, F. J. (2023), A primal–dual partial inverse algorithm for constrained monotone inclusions: Applications to stochastic programming and mean field games, Appl. Math. Optim. 87, art. 21.10.1007/s00245-022-09921-9CrossRefGoogle Scholar
Briceño-Arias, L. M., Kalise, D. and Silva, F. J. (2018), Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim. 56, 801–836.10.1137/16M1095615CrossRefGoogle Scholar
Brogliato, B. and Tanwani, A. (2020), Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Rev. 62, 3–129.10.1137/18M1234795CrossRefGoogle Scholar
Brogliato, B., Lozano, R., Maschke, B. and Egeland, O. (2007), Dissipative Systems Analysis and Control: Theory and Applications, second edition, Springer.10.1007/978-1-84628-517-2CrossRefGoogle Scholar
Browder, F. E. (1963), The solvability of non-linear functional equations, Duke Math. J. 30, 557–566.10.1215/S0012-7094-63-03061-8CrossRefGoogle Scholar
Browder, F. E. (1965), Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 118, 338–351.10.1090/S0002-9947-1965-0180884-9CrossRefGoogle Scholar
Browder, F. E. (1968/1976), Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math. 18, 1–308.Google Scholar
Bruck, R. E. (1973), The iterative solution of the equation y ∈ x+Tx for a monotone operator T in Hilbert space, Bull. Amer. Math. Soc. 79, 1258–1261.10.1090/S0002-9904-1973-13404-4CrossRefGoogle Scholar
Bruck, R. E. (1974), A strongly convergent iterative solution of 0 ∈ U(x) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl. 48, 114–126.10.1016/0022-247X(74)90219-4CrossRefGoogle Scholar
Bruck, R. E. (1975), An iterative solution of a variational inequality for certain monotone operators in Hilbert space, Bull. Amer. Math. Soc. 81, 890–892. Corrigendum: 82, 353 (1976).10.1090/S0002-9904-1975-13874-2CrossRefGoogle Scholar
Buck, R. C. (1956), Advanced Calculus, first edition, McGraw-Hill.Google Scholar
Bùi, M. N. (2022a), A decomposition method for solving multicommodity network equilibria, Oper. Res. Lett. 50, 40–44.10.1016/j.orl.2021.12.002CrossRefGoogle Scholar
Bùi, M. N. (2022b), Projective splitting as a warped proximal algorithm, Appl. Math. Optim. 85, art. 4.10.1007/s00245-022-09868-xCrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L. (2020a), The Douglas–Rachford algorithm converges only weakly, SIAM J. Control Optim. 58, 1118–1120.10.1137/19M1308451CrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L. (2020b), Warped proximal iterations for monotone inclusions, J. Math. Anal. Appl. 491, art. 124315.10.1016/j.jmaa.2020.124315CrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L. (2021), Bregman forward–backward operator splitting, Set-Valued Var. .Anal 29, 583–603.10.1007/s11228-020-00563-zCrossRefGoogle Scholar
Bùi, M. N. and Combettes, P. L. (2022a), Analysis and numerical solution of a modular convex Nash equilibrium problem, J. Convex Anal. 29, 1007–1021.Google Scholar
Bùi, M. N. and Combettes, P. L. (2022b), Multivariate monotone inclusions in saddle form, Math. Oper. Res. 47, 1082–1109.10.1287/moor.2021.1161CrossRefGoogle Scholar
Bùi, M. N., Combettes, P. L. and Woodstock, Z. C. (2022), Block-activated algorithms for multicomponent fully nonsmooth minimization, in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2022), pp. 5428–5432.Google Scholar
Byrne, C., Censor, Y., Gibali, A. and Reich, S. (2012), The split common null point problem, J. Nonlinear Convex Anal. 13, 759–775.Google Scholar
Camlibel, M. K. and Schumacher, J. M. (2016), Linear passive systems and maximal monotone mappings, Math. Program. B157, 397–420.10.1007/s10107-015-0945-7CrossRefGoogle Scholar
Cauchy, A. (1847), Méthode générale pour la résolution des systèmes d’équations simultanées, C. R. Acad. Sci. Paris 25, 536–538.Google Scholar
Cederbaum, I. (1962), On optimal operation of communication nets, J. Franklin Inst. 274, 130–141.10.1016/0016-0032(62)90401-5CrossRefGoogle Scholar
Cegielski, A. (2012), Iterative Methods for Fixed Point Problems in Hilbert Spaces, Vol. 2057 of Lecture Notes in Mathematics, Springer.Google Scholar
Censor, Y. and Zaknoon, M. (2018), Algorithms and convergence results of projection methods for inconsistent feasibility problems: A review, Pure Appl. Funct. Anal. 3, 565–586.Google Scholar
Censor, Y. and Zenios, S. A. (1992), Proximal minimization algorithm with D-functions, J. Optim. Theory Appl. 73, 451–464.10.1007/BF00940051CrossRefGoogle Scholar
Chaffey, T. and Sepulchre, R. (2024), Monotone one-port circuits, IEEE Trans. Automat. Control 69, 783–796.10.1109/TAC.2023.3274690CrossRefGoogle Scholar
Chaffey, T., Banert, S., Giselsson, P. and Pates, R. (2023a), Circuit analysis using monotone+skew splitting, Eur. J. Control 74, art. 100854.10.1016/j.ejcon.2023.100854CrossRefGoogle Scholar
Chaffey, T., Forni, F. and Sepulchre, R. (2023b), Graphical nonlinear system analysis, IEEE Trans. Automat. Control 68, 6067–6081.10.1109/TAC.2023.3234016CrossRefGoogle Scholar
Chambolle, A. (2004), An algorithm for total variation minimization and applications, J. Math. Imaging Vision 20, 89–97.Google Scholar
Chambolle, A. (2005), Total variation minimization and a class of binary MRF models, in Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2005), Vol. 3757 of Lecture Notes in Computer Science, Springer, pp. 136–152.10.1007/11585978_10CrossRefGoogle Scholar
Chambolle, A. and Dossal, C. (2015), On the convergence of the iterates of the ‘fast iterative shrinkage/thresholding algorithm’, J. Optim. Theory Appl. 166, 968–982.10.1007/s10957-015-0746-4CrossRefGoogle Scholar
Chambolle, A. and Pock, T. (2011), A first-order primal–dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision 40, 120–145.10.1007/s10851-010-0251-1CrossRefGoogle Scholar
Chambolle, A. and Pock, T. (2016), An introduction to continuous optimization for imaging, Acta Numer. 25, 161–319.10.1017/S096249291600009XCrossRefGoogle Scholar
Chan, R. H., Setzer, S. and Steidl, G. (2008), Inpainting by flexible Haar-wavelet shrinkage, SIAM J. Imaging Sci. 1, 273–293.10.1137/070711499CrossRefGoogle Scholar
Chan, S. H., Wang, X. and Elgendy, O. A. (2017), Plug-and-play ADMM for image restoration: Fixed-point convergence and applications, IEEE Trans. Comput. Imaging 3, 84–98.10.1109/TCI.2016.2629286CrossRefGoogle Scholar
Chaux, C., El-Gheche, M., Farah, J., Pesquet, J.-C. and Pesquet-Popescu, B. (2013), A parallel proximal splitting method for disparity estimation from multicomponent images under illumination variation, J. Math. Imaging Vision 47, 167–178.10.1007/s10851-012-0361-zCrossRefGoogle Scholar
Chen, G. and Teboulle, M. (1994), A proximal-based decomposition method for convex minimization problems, Math. Program. 64, 81–101.10.1007/BF01582566CrossRefGoogle Scholar
Chen, G. H.-G. and Rockafellar, R. T. (1997), Convergence rates in forward–backward splitting, SIAM J. Optim. 7, 421–444.10.1137/S1052623495290179CrossRefGoogle Scholar
Cheney, E. W. and Goldstein, A. A. (1959a), Newton’s method for convex programming and Tchebycheff approximation, Numer. Math. 1, 253–268.10.1007/BF01386389CrossRefGoogle Scholar
Cheney, W. and Goldstein, A. A. (1959b), Proximity maps for convex sets, Proc. Amer. Math. Soc. 10, 448–450.10.1090/S0002-9939-1959-0105008-8CrossRefGoogle Scholar
Chierchia, G., Chouzenoux, E., Combettes, P. L. and Pesquet, J.-C. (2016), The proximity operator repository. Available at http://proximity-operator.net/.Google Scholar
Clason, C. and Valkonen, T. (2017), Primal–dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization, SIAM J. Optim. 27, 1314–1339.10.1137/16M1080859CrossRefGoogle Scholar
Cohen, G. (1987), Nash equilibria: Gradient and decomposition algorithms, Large Scale Syst. 12, 173–184.Google Scholar
Combettes, P. L. (2001a), Fejér-monotonicity in convex optimization, in Encyclopedia of Optimization (Floudas, C. A. and Pardalos, P. M., eds), Vol. 2, Springer, pp. 106–114. Also available in second edition (2009), pp. 1016–1024.Google Scholar
Combettes, P. L. (2001b), Quasi-Fejérian analysis of some optimization algorithms, in Inherently Parallel Algorithms for Feasibility and Optimization (Butnariu, D., Censor, Y. and Reich, S., eds), Elsevier, pp. 115–152.10.1016/S1570-579X(01)80010-0CrossRefGoogle Scholar
Combettes, P. L. (2004), Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization 53, 475–504.10.1080/02331930412331327157CrossRefGoogle Scholar
Combettes, P. L. (2009), Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal. 16, 727–748.Google Scholar
Combettes, P. L. (2013a), Can one genuinely split m > 2 monotone operators?, in Workshop on Algorithms and Dynamics for Games and Optimization (ADGO 2013). Available at https://pcombet.math.ncsu.edu/2013open-pbs1.pdf.+2+monotone+operators?,+in+Workshop+on+Algorithms+and+Dynamics+for+Games+and+Optimization+(ADGO+2013).+Available+at+https://pcombet.math.ncsu.edu/2013open-pbs1.pdf.>Google Scholar
Combettes, P. L. (2013b), Systems of structured monotone inclusions: Duality, algorithms, and applications, SIAM J. Optim. 23, 2420–2447.10.1137/130904160CrossRefGoogle Scholar
Combettes, P. L. (2018), Monotone operator theory in convex optimization, Math. Program. B170, 177–206.10.1007/s10107-018-1303-3CrossRefGoogle Scholar
Combettes, P. L. (2023), Resolvent and proximal compositions, Set-Valued Var. Anal. 31, art. 22.10.1007/s11228-023-00678-zCrossRefGoogle Scholar
Combettes, P. L. and Bondon, P. (1999), Hard-constrained inconsistent signal feasibility problems, IEEE Trans. Signal Process. 47, 2460–2468.10.1109/78.782189CrossRefGoogle Scholar
Combettes, P. L. and Eckstein, J. (2018), Asynchronous block-iterative primal–dual decomposition methods for monotone inclusions, Math. Program. B168, 645–672.10.1007/s10107-016-1044-0CrossRefGoogle Scholar
Combettes, P. L. and Glaudin, L. E. (2017), Quasinonexpansive iterations on the affine hull of orbits: From Mann’s mean value algorithm to inertial methods, SIAM J. Optim. 27, 2356–2380.10.1137/17M112806XCrossRefGoogle Scholar
Combettes, P. L. and Glaudin, L. E. (2019), Proximal activation of smooth functions in splitting algorithms for convex image recovery, SIAM J. Imaging Sci. 12, 1905–1935.10.1137/18M1224763CrossRefGoogle Scholar
Combettes, P. L. and Glaudin, L. E. (2021), Solving composite fixed point problems with block updates, Adv. Nonlinear Anal. 10, 1154–1177.10.1515/anona-2020-0173CrossRefGoogle Scholar
Combettes, P. L. and Hirstoaga, S. A. (2005), Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6, 117–136.Google Scholar
Combettes, P. L. and Müller, C. L. (2020), Perspective maximum likelihood-type estimation via proximal decomposition, Electron. J. Statist. 14, 207–238.10.1214/19-EJS1662CrossRefGoogle Scholar
Combettes, P. L. and Müller, C. L. (2021), Regression models for compositional data: General log-contrast formulations, proximal optimization, and microbiome data applications, Statist. Biosci. 13, 217–242.10.1007/s12561-020-09283-2CrossRefGoogle Scholar
Combettes, P. L. and Pesquet, J.-C. (2007), A Douglas–Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE J. Select. Topics Signal Process. 1, 564–574.10.1109/JSTSP.2007.910264CrossRefGoogle Scholar
Combettes, P. L. and Pesquet, J.-C. (2011), Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Vol. 49 of Springer Optimization and its Applications, Springer, pp. 185–212.Google Scholar
Combettes, P. L. and Pesquet, J.-C. (2012), Primal–dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal. 20, 307–330.10.1007/s11228-011-0191-yCrossRefGoogle Scholar
Combettes, P. L. and Pesquet, J.-C. (2015), Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping, SIAM J. Optim. 25, 1221–1248.10.1137/140971233CrossRefGoogle Scholar
Combettes, P. L. and Pesquet, J.-C. (2020), Deep neural network structures solving variational inequalities,.Set-Valued Var. Anal 28, 491–518.10.1007/s11228-019-00526-zCrossRefGoogle Scholar
Combettes, P. L. and Pesquet, J.-C. (2021), Fixed point strategies in data science, IEEE Trans. Signal Process. 69, 3878–3905.10.1109/TSP.2021.3069677CrossRefGoogle Scholar
Combettes, P. L. and Vũ, B. C. (2013), Variable metric quasi-Fejér monotonicity, Nonlinear Anal. 78, 17–31.10.1016/j.na.2012.09.008CrossRefGoogle Scholar
Combettes, P. L. and Vũ, B. C. (2014), Variable metric forward–backward splitting with applications to monotone inclusions in duality, Optimization 63, 1289–1318.10.1080/02331934.2012.733883CrossRefGoogle Scholar
Combettes, P. L. and Wajs, V. R. (2005), Signal recovery by proximal forward–backward splitting, Multiscale Model. Simul. 4, 1168–1200.10.1137/050626090CrossRefGoogle Scholar
Combettes, P. L. and Woodstock, Z. C. (2022), A variational inequality model for the construction of signals from inconsistent nonlinear equations, SIAM J. Imaging Sci. 15, 84–109.10.1137/21M1420368CrossRefGoogle Scholar
Combettes, P. L. and Yamada, I. (2015), Compositions and convex combinations of averaged nonexpansive operators, J. Math. Anal. Appl. 425, 55–70.10.1016/j.jmaa.2014.11.044CrossRefGoogle Scholar
Combettes, P. L., Dũng, Ðinh and Vũ, B. C. (2010), Dualization of signal recovery problems, Set-Valued Var. Anal. 18, 373–404.10.1007/s11228-010-0147-7CrossRefGoogle Scholar
Combettes, P. L., Dũng, Ðinh and Vũ, B. C. (2011), Proximity for sums of composite functions, J. Math. Anal. Appl. 380, 680–688.10.1016/j.jmaa.2011.02.079CrossRefGoogle Scholar
Combettes, P. L., Salzo, S. and Villa, S. (2018), Consistent learning by composite proximal thresholding, Math. Program. B167, 99–127.10.1007/s10107-017-1133-8CrossRefGoogle Scholar
Condat, L. (2013), A primal–dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl. 158, 460–479.10.1007/s10957-012-0245-9CrossRefGoogle Scholar
Condat, L., Kitahara, D., Contreras, A. and Hirabayashi, A. (2023), Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists, SIAM Rev. 65, 375–435.10.1137/20M1379344CrossRefGoogle Scholar
Curry, H. B. (1944), The method of steepest descent for non-linear minimization problems, Quart. Appl. Math. 2, 258–261.10.1090/qam/10667CrossRefGoogle Scholar
Dafermos, S. (1980), Traffic equilibrium and variational inequalities, Transport. Sci. 14, 42–54.10.1287/trsc.14.1.42CrossRefGoogle Scholar
Darboux, G. (1875), Mémoire sur les fonctions discontinues, Ann. Sci. École Normale Sup. 4, 57–112.10.24033/asens.122CrossRefGoogle Scholar
Daubechies, I., Defrise, M. and De Mol, C. (2004), An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57, 1413–1457.10.1002/cpa.20042CrossRefGoogle Scholar
Davis, D. and Yin, W. (2017), A three-operator splitting scheme and its optimization applications, Set-Valued Var. Anal. 25, 829–858.10.1007/s11228-017-0421-zCrossRefGoogle Scholar
Desoer, C. A. and Vidyasagar, M. (1975), Feedback Systems: Input–Output Properties, Academic.Google Scholar
Desoer, C. A. and Wu, F. F. (1974), Nonlinear monotone networks, SIAM J. Appl. Math. 26, 315–333.10.1137/0126030CrossRefGoogle Scholar
Dexter, N., Tran, H. and Webster, C. G. (2022), On the strong convergence of forward–backward splitting in reconstructing jointly sparse signals, Set-Valued Var. Anal. 30, 543–557.10.1007/s11228-021-00603-2CrossRefGoogle Scholar
Doležal, V. (1979a), Feedback systems described by monotone operators, SIAM J. Control Optim. 17, 339–364.10.1137/0317027CrossRefGoogle Scholar
Doležal, V. (1979b), Monotone Operators and Applications in Control and Network Theory, Elsevier.Google Scholar
Dong, Q.-L., Cho, Y. J., He, S., Pardalos, P. M. and Rassias, T. M. (2022), The Krasnosel’skiĭ–Mann Iterative Method: Recent Progress and Applications, Springer.10.1007/978-3-030-91654-1CrossRefGoogle Scholar
Dong, Y. (2005), An LS-free splitting method for composite mappings, Appl. Math. Lett. 18, 843–848.10.1016/j.aml.2004.09.013CrossRefGoogle Scholar
Douglas, J. (1955), On the numerical integration of ${\partial}^2u/\partial {x}^2+{\partial}^2u/\partial {y}^2=\partial u/\partial t$ by implicit methods, J. Soc. Indust. Appl. Math. 3, 42–65.10.1137/0103004CrossRefGoogle Scholar
Douglas, J. and Rachford, H. H. (1956), On the numerical solution of heat conduction problems in two or three space variables, Trans. Amer. Math. Soc. 82, 421–439.10.1090/S0002-9947-1956-0084194-4CrossRefGoogle Scholar
Duffin, R. J. (1946), Nonlinear networks I, Bull. Amer. Math. Soc. 52, 833–838.10.1090/S0002-9904-1946-08650-4CrossRefGoogle Scholar
Duffin, R. J. (1947), Nonlinear networks IIa, Bull. Amer. Math. Soc. 53, 963–971.10.1090/S0002-9904-1947-08917-5CrossRefGoogle Scholar
Duffin, R. J. (1948), Nonlinear networks IIb, Bull. Amer. Math. Soc. 54, 119–127.10.1090/S0002-9904-1948-08964-9CrossRefGoogle Scholar
Dykstra, R. L. (1983), An algorithm for restricted least squares regression, J. Amer. Statist. Assoc. 78, 837–842.10.1080/01621459.1983.10477029CrossRefGoogle Scholar
Eaves, B. C. (1984), Subdivisions from primal and dual cones and polytopes, Linear Algebra Appl. 62, 277–285.10.1016/0024-3795(84)90103-4CrossRefGoogle Scholar
Eckstein, J. (1993), Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming, Math. Oper. Res. 18, 202–226.10.1287/moor.18.1.202CrossRefGoogle Scholar
Eckstein, J. (1994), Some saddle-function splitting methods for convex programming, Optim. Methods Softw. 4, 75–83.10.1080/10556789408805578CrossRefGoogle Scholar
Eckstein, J. (2017), A simplified form of block-iterative operator splitting and an asynchronous algorithm resembling the multi-block alternating direction method of multipliers, J. Optim. Theory Appl. 173, 155–182.10.1007/s10957-017-1074-7CrossRefGoogle Scholar
Eckstein, J. and Bertsekas, D. P. (1992), On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program. 55, 293–318.10.1007/BF01581204CrossRefGoogle Scholar
Eckstein, J. and Ferris, M. C. (1999), Smooth methods of multipliers for complementarity problems, Math. Program. A86, 65–90.10.1007/s101079900076CrossRefGoogle Scholar
Eckstein, J. and Svaiter, B. F. (2008), A family of projective splitting methods for the sum of two maximal monotone operators, Math. Program. 111, 173–199.10.1007/s10107-006-0070-8CrossRefGoogle Scholar
Eckstein, J. and Svaiter, B. F. (2009), General projective splitting methods for sums of maximal monotone operators, SIAM J. Control Optim. 48, 787–811.10.1137/070698816CrossRefGoogle Scholar
Eckstein, J., Watson, J.-P. and Woodruff, D. L. (2023), Projective hedging algorithms for multistage stochastic programming, supporting distributed and asynchronous implementation, Oper. Res. Available at doi:10.1287/opre.2022.0228.CrossRefGoogle Scholar
Eicke, B. (1992), Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numer. Funct. Anal. Optim. 13, 413–429.10.1080/01630569208816489CrossRefGoogle Scholar
Ekeland, I. and Temam, R. (1974), Analyse Convexe et Problèmes Variationnels, Dunod. English translation: Convex Analysis and Variational Problems, SIAM (1999).Google Scholar
Eremin, I. I. (1968a), Methods of Fejér approximations in convex programming, Mat. Zametki 3, 217–234.Google Scholar
Eremin, I. I. (1968b), On the speed of convergence in the method of Fejér approximations, Mat. Zametki 4, 53–62.Google Scholar
Ermol’ev, Y. M. and Tuniev, A. D. (1968), Random Fejér and quasi-Fejér sequences, Theory of Optimal Solutions – Akad. Nauk Ukrainskoĭ SSR Kiev 2, 76–83.Google Scholar
Facchinei, F. and Pang, J.-S. (2003), Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer.Google Scholar
Facchinei, F., Fischer, A. and Piccialli, V. (2007), On generalized Nash games and variational inequalities, Oper. Res. Lett. 35, 159–164.10.1016/j.orl.2006.03.004CrossRefGoogle Scholar
Fejér, L. (1922), Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Math. Ann. 85, 41–48.10.1007/BF01449600CrossRefGoogle Scholar
Fichera, G. (1963), Sul problema elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Ser. VIII 34, 138–142.Google Scholar
Figueiredo, M. A. T. and Nowak, R. D. (2003), An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process. 12, 906–916.10.1109/TIP.2003.814255CrossRefGoogle Scholar
Fortin, M. and Glowinski, R., eds (1983), Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland.Google Scholar
Froda, A. (1929), Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles, Hermann.Google Scholar
Fukushima, M. (1996), The primal Douglas–Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem, Math. Program. 72, 1–15.10.1007/BF02592328CrossRefGoogle Scholar
Gabay, D. (1983), Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (Fortin, M. and Glowinski, R., eds), North-Holland, pp. 299–331.10.1016/S0168-2024(08)70034-1CrossRefGoogle Scholar
Gabay, D. and Mercier, B. (1976), A dual algorithm for the solution of nonlinear variational problems via finite elements approximations, Comput. Math. Appl. 2, 17–40.10.1016/0898-1221(76)90003-1CrossRefGoogle Scholar
Gaffke, N. and Mathar, R. (1989), A cyclic projection algorithm via duality, Metrika 36, 29–54.10.1007/BF02614077CrossRefGoogle Scholar
Gandy, S., Recht, B. and Yamada, I. (2011), Tensor completion and low-n-rank tensor recovery via convex optimization, Inverse Problems 27, art. 025010.10.1088/0266-5611/27/2/025010CrossRefGoogle Scholar
Garrigos, G., Rosasco, L. and Villa, S. (2023), Convergence of the forward–backward algorithm: Beyond the worst-case with the help of geometry, Math. Program. A198, 937–996.10.1007/s10107-022-01809-4CrossRefGoogle Scholar
Gauss, C. F. (1809), Theoria Motus Corporum Coelestium, Perthes and Besser.Google Scholar
Gautam, P., Sahu, D. R., Dixit, A. and Som, T. (2021), Forward–backward-half forward dynamical systems for monotone inclusion problems with application to v-GNE, J. Optim. Theory Appl. 190, 491–523.10.1007/s10957-021-01891-2CrossRefGoogle Scholar
Genel, A. and Lindenstrauss, J. (1975), An example concerning fixed points, Israel J. Math. 22, 81–86.10.1007/BF02757276CrossRefGoogle Scholar
Ghizzetti, A., ed. (1969), Theory and Applications of Monotone Operators, Edizioni Oderisi.Google Scholar
Ghoussoub, N. (2009), Self-Dual Partial Differential Systems and Their Variational Principles, Springer.Google Scholar
Giselsson, P. (2021), Nonlinear forward–backward splitting with projection correction, SIAM J. Optim. 31, 2199–2226.10.1137/20M1345062CrossRefGoogle Scholar
Glowinski, R. and Le Tallec, P., eds (1989), Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM.10.1137/1.9781611970838CrossRefGoogle Scholar
Glowinski, R. and Marrocco, A. (1974), Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires, C. R. Acad. Sci. Paris A278, 1649–1652. See also RAIRO Anal. Numer. 9, 41–76 (1975).Google Scholar
Glowinski, R., Osher, S. J. and Yin, W., eds (2016), Splitting Methods in Communication, Imaging, Science, and Engineering, Springer.10.1007/978-3-319-41589-5CrossRefGoogle Scholar
Goeleven, D. (2017), Complementarity and Variational Inequalities in Electronics, Academic.Google Scholar
Goldburg, M. and Marks, R. J. II (1985), Signal synthesis in the presence of an inconsistent set of constraints, IEEE Trans. Circuits Syst. 32, 647–663.10.1109/TCS.1985.1085777CrossRefGoogle Scholar
Goldstein, A. A. (1964), Convex programming in Hilbert space, Bull. Amer. Math. Soc. 70, 709–710.10.1090/S0002-9904-1964-11178-2CrossRefGoogle Scholar
Golomb, M. (1935), Zur Theorie der nichtlinearen Integralgleichungen, Integralgleichungssysteme und allgemeinen Funktionalgleichungen, Math. Z. 39, 45–75.10.1007/BF01201344CrossRefGoogle Scholar
Golomb, M. (1936), Über Systeme von nichtlinearen Integralgleichungen, Publ. Math. Univ. Belgrade 5, 52–83.Google Scholar
Gol’shtein, E. G. (1987), A general approach to decomposition of optimization systems, Sov. J. Comput. Syst. Sci. 25, 105–114.Google Scholar
Gol’shtein, E. G. and Tret’yakov, N. V. (1979), Modified Lagrangians in convex programming and their generalizations, Math. Program. Studies 10, 86–97.10.1007/BFb0120845CrossRefGoogle Scholar
Gol’shtein, E. G. and Tret’yakov, N. V. (1996), Modified Lagrangians and Monotone Maps in Optimization, Wiley.Google Scholar
Groetsch, C. W. (1972), A note on segmenting Mann iterates, J. Math. Anal. Appl. 40, 369–372.10.1016/0022-247X(72)90056-XCrossRefGoogle Scholar
Gubin, L. G., Polyak, B. T. and Raik, E. V. (1967), The method of projections for finding the common point of convex sets, Comput. Math. Math. Phys. 7, 1–24.10.1016/0041-5553(67)90113-9CrossRefGoogle Scholar
Güler, O. (1991), On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29, 403–419.10.1137/0329022CrossRefGoogle Scholar
Hahn, H. (1921), Theorie der reellen Funktionen, Springer.10.1007/978-3-642-52624-4CrossRefGoogle Scholar
Han, S. P. (1988), A successive projection method, Math. Program. 40, 1–14.10.1007/BF01580719CrossRefGoogle Scholar
Haugazeau, Y. (1967), Sur la minimisation de formes quadratiques avec contraintes, C. R. Acad. Sci. Paris A264, 322–324.Google Scholar
Haugazeau, Y. (1968), Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes. Thesis, Université de Paris.Google Scholar
He, B. and Yuan, X. (2012), Convergence analysis of primal–dual algorithms for a saddle-point problem: From contraction perspective, SIAM J. Imaging Sci. 5, 119–149.10.1137/100814494CrossRefGoogle Scholar
He, Y. and Monteiro, R. D. C. (2015), Accelerating block-decomposition first-order methods for solving composite saddle-point and two-player Nash equilibrium problems, SIAM J. Optim. 25, 2182–2211.10.1137/130943649CrossRefGoogle Scholar
Hestenes, M. R. (1969), Multiplier and gradient methods, J. Optim. Theory Appl. 4, 303–320.10.1007/BF00927673CrossRefGoogle Scholar
Hu, H., Sotirov, R. and Wolkowicz, H. (2023), Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs, Math. Program. A200, 475–529.10.1007/s10107-022-01890-9CrossRefGoogle Scholar
Hundal, H. S. (2004), An alternating projection that does not converge in norm, Nonlinear Anal. 57, 35–61.10.1016/j.na.2003.11.004CrossRefGoogle Scholar
Idrissi, H., Lefebvre, O. and Michelot, C. (1989), Applications and numerical convergence of the partial inverse method, in Optimization, Vol. 1405 of Lecture Notes in Mathematics, Springer, pp. 39–54.10.1007/BFb0083585CrossRefGoogle Scholar
Jenatton, R., Mairal, J., Obozinski, G. and Bach, F. (2011), Proximal methods for hierarchical sparse coding, J. Mach. Learn. Res. 12, 2297–2334.Google Scholar
Jensen, J. L. W. V. (1906), Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. 30, 175–193.10.1007/BF02418571CrossRefGoogle Scholar
Johnstone, P. R. and Eckstein, J. (2019), Convergence rates for projective splitting, SIAM J. Optim. 29, 1931–1957.10.1137/18M1203523CrossRefGoogle Scholar
Johnstone, P. R. and Eckstein, J. (2020), Projective splitting with forward steps only requires continuity, Optim. Lett. 14, 229–247.10.1007/s11590-019-01509-7CrossRefGoogle Scholar
Johnstone, P. R. and Eckstein, J. (2021), Single-forward-step projective splitting: Exploiting cocoercivity, Comput. Optim. Appl. 78, 125–166.10.1007/s10589-020-00238-3CrossRefGoogle Scholar
Johnstone, P. R. and Eckstein, J. (2022), Projective splitting with forward steps, Math. Program. A191, 631–670.10.1007/s10107-020-01565-3CrossRefGoogle Scholar
Joly, J. L. and Laurent, P. J. (1971), Stability and duality in convex minimization problems, Rev. Fr. Inform. Rech. Opér. 5, 3–42.Google Scholar
Kačurovskiĭ, R. I. (1960), Monotone operators and convex functionals, Uspekhi Mat. Nauk 15, 213–215.Google Scholar
Kačurovskiĭ, R. I. (1968), Non-linear monotone operators in Banach spaces, Russian Math. Surveys 23, 117–165.10.1070/RM1968v023n02ABEH001239CrossRefGoogle Scholar
Kato, T. (1980), Perturbation Theory for Linear Operators, second edition, Springer.Google Scholar
Kelley, J. E. (1960), The cutting-plane method for solving convex programs, J. Soc Indust. Appl. Math. 8, 703–712.10.1137/0108053CrossRefGoogle Scholar
Kellogg, R. B. (1969), A nonlinear alternating direction method, Math. Comp. 23, 23–27.10.1090/S0025-5718-1969-0238507-3CrossRefGoogle Scholar
Kinderlehrer, D. and Stampacchia, G. (1980), An Introduction to Variational Inequalities and Their Applications, Academic.Google Scholar
Kōmura, Y. (1967), Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan 19, 493–507.10.2969/jmsj/01940493CrossRefGoogle Scholar
Korpelevič, G. M. (1976), The extragradient method for finding saddle points and other problems, Èkonom. i Mat. Metody 12, 747–756.Google Scholar
Krasnosel’skiĭ, M. A. (1955), Two remarks on the method of successive approximations, Uspekhi Mat. Nauk 10, 123–127.Google Scholar
Kryanev, A. V. (1973), The solution of incorrectly posed problems by methods of successive approximations, Soviet Math. Dokl. 14, 673–676.Google Scholar
Latafat, P. and Patrinos, P. (2017), Asymmetric forward–backward-adjoint splitting for solving monotone inclusions involving three operators, Comput. Optim. Appl. 68, 57–93.10.1007/s10589-017-9909-6CrossRefGoogle Scholar
Laurent, P. J. and Martinet, B. (1970), Méthodes duales pour le calcul du minimum d’une fonction convexe sur une intersection de convexes, in Symposium on Optimization, Vol. 132 of Lecture Notes in Mathematics, Springer, pp. 159–180.10.1007/BFb0066681CrossRefGoogle Scholar
Lawrence, J. and Spingarn, J. E. (1987), On fixed points of non-expansive piecewise isometric mappings, Proc. London Math. Soc. 55, 605–624.10.1112/plms/s3-55.3.605CrossRefGoogle Scholar
Legendre, A. M. (1805), Nouvelles Méthodes pour la Détermination des Orbites des Comètes, Firmin Didot.Google Scholar
Lemaire, B. (1989), The proximal algorithm, in New Methods in Optimization and Their Industrial Uses (Penot, J. P., ed.), Vol. 87 of International Series of Numerical Mathematics, Birkhäuser, pp. 73–87.Google Scholar
Lemaire, B. (1996), Stability of the iteration method for nonexpansive mappings, Serdica Math. J. 22, 331–340.Google Scholar
Lemaire, B. (1997), Which fixed point does the iteration method select?, in Recent Advances in Optimization, Vol. 452 of Lecture Notes in Economics and Mathematical Systems, Springer, pp. 154–167.10.1007/978-3-642-59073-3_11CrossRefGoogle Scholar
Lenoir, A. and Mahey, P. (2017), A survey on operator splitting and decomposition of convex programs, RAIRO-Oper. Res. 51, 17–41.10.1051/ro/2015065CrossRefGoogle Scholar
Leray, J. and Lions, J.-L. (1965), Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder, Bull. Soc. Math. France 93, 97–107.10.24033/bsmf.1617CrossRefGoogle Scholar
Levitin, E. S. and Polyak, B. T. (1966), Constrained minimization methods, USSR Comput. Math. Math. Phys. 6, 1–50.10.1016/0041-5553(66)90114-5CrossRefGoogle Scholar
Lieutaud, J. (1969a), Approximations d’opérateurs monotones par des méthodes de splitting, in Theory and Applications of Monotone Operators (Ghizzetti, A., ed.), Edizioni Oderisi, pp. 259–264.Google Scholar
Lieutaud, J. (1969b), Approximation d’opérateurs par des méthodes de décomposition. Thesis, Université de Paris.Google Scholar
Lindstrom, S. and Sims, B. (2021), Survey: Sixty years of Douglas–Rachford, J. Aust. Math. Soc. 110, 333–370.10.1017/S1446788719000570CrossRefGoogle Scholar
Lions, J.-L. (1969), Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod.Google Scholar
Lions, J.-L., ed. (2010), Numerical Analysis of Partial Differential Equations, Springer.10.1007/978-3-642-11057-3CrossRefGoogle Scholar
Lions, P.-L. (1978), Une méthode itérative de résolution d’une inéquation variationnelle, Israel J. Math. 31, 204–208.10.1007/BF02760552CrossRefGoogle Scholar
Lions, P.-L. and Mercier, B. (1979), Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16, 964–979.10.1137/0716071CrossRefGoogle Scholar
Liu, J. and Wright, S. J. (2015), Asynchronous stochastic coordinate descent: Parallelism and convergence properties, SIAM J. Optim. 25, 351–376.10.1137/140961134CrossRefGoogle Scholar
Lu, H., Freund, R. M. and Nesterov, Y. (2018), Relatively smooth convex optimization by first-order methods, and applications, SIAM J. Optim. 28, 333–354.10.1137/16M1099546CrossRefGoogle Scholar
Machado, M. P. (2018), On the complexity of the projective splitting and Spingarn’s methods for the sum of two maximal monotone operators, J. Optim. Theory Appl. 178, 153–190.10.1007/s10957-018-1310-9CrossRefGoogle Scholar
Machado, P. and Sicre, M. R. (2023), A projective splitting method for monotone inclusions: Iteration-complexity and application to composite optimization, J. Optim. Theory Appl. 198, 552–587.10.1007/s10957-023-02214-3CrossRefGoogle Scholar
Mahey, P. and Pham, D. T. (1993), Partial regularization of the sum of two maximal monotone operators, RAIRO Modélisation Math. Analyse Numér. 27, 375–392.10.1051/m2an/1993270303751CrossRefGoogle Scholar
Mahey, P., Oualibouch, S. and Tao, P. Dinh (1995), Proximal decomposition on the graph of a maximal monotone operator, SIAM J. Optim. 5, 454–466.10.1137/0805023CrossRefGoogle Scholar
Malitsky, Y. and Tam, M. K. (2023), Resolvent splitting for sums of monotone operators with minimal lifting, Math. Program. A201, 231–262.10.1007/s10107-022-01906-4CrossRefGoogle Scholar
Mann, W. R. (1953), Mean value methods in iteration, Proc. Amer. Math. Soc. 4, 506–510.10.1090/S0002-9939-1953-0054846-3CrossRefGoogle Scholar
Martinet, B. (1970), Régularisation d’inéquations variationnelles par approximations successives, Rev. Fr. Inform. Rech. Opér. 4, 154–158.Google Scholar
Martinet, B. (1972), Détermination approchée d’un point fixe d’une application pseudo-contractante: Cas de l’application prox, C. R. Acad. Sci. Paris A274, 163–165.Google Scholar
Martínez-Legaz, J. E. and Seeger, A. (1994), A general cone decomposition theory based on efficiency, Math. Program. 65, 1–20.10.1007/BF01581687CrossRefGoogle Scholar
Mercier, B. (1979), Topics in Finite Element Solution of Elliptic Problems, Vol. 63 of Lectures on Mathematics, Tata Institute of Fundamental Research, Bombay.Google Scholar
Mercier, B. (1980), Inéquations Variationnelles de la Mécanique, Vol. 80.01 of Publications Mathématiques d’Orsay, Université de Paris-XI, Orsay, France.Google Scholar
Micchelli, C. A., Morales, J. M. and Pontil, M. (2013), Regularizers for structured sparsity, Adv. Comput. Math. 38, 455–489.10.1007/s10444-011-9245-9CrossRefGoogle Scholar
Millar, W. (1951), Some general theorems for non-linear systems possessing resistance, London Edin. Dublin Philos. Mag. 42, 1150–1160.10.1080/14786445108561361CrossRefGoogle Scholar
Minty, G. J. (1960), Monotone networks, Proc. R. Soc. Lond. A 57, 194–212.Google Scholar
Minty, G. J. (1961), Solving steady-state nonlinear networks of ‘monotone’ elements, IRE Trans. Circuit Theory 8, 99–104.10.1109/TCT.1961.1086765CrossRefGoogle Scholar
Minty, G. J. (1962), Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29, 341–346.10.1215/S0012-7094-62-02933-2CrossRefGoogle Scholar
Minty, G. J. (1963), On a ‘monotonicity’ method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. USA 50, 1038–1041.10.1073/pnas.50.6.1038CrossRefGoogle ScholarPubMed
Minty, G. J. (1964), On the monotonicity of the gradient of a convex function, Pacific J. Math. 14, 243–247.10.2140/pjm.1964.14.243CrossRefGoogle Scholar
Minty, G. J. (1969), On some aspects of the theory of monotone operators, in Theory and Applications of Monotone Operators (Ghizzetti, A., ed.), Edizioni Oderisi, pp. 67–82.Google Scholar
Mishchenko, K., Iutzeler, F. and Malick, J. (2020), A distributed flexible delay-tolerant proximal gradient algorithm, SIAM J. Optim. 30, 933–959.10.1137/18M1194699CrossRefGoogle Scholar
Mizoguchi, T. and Yamada, I. (2019), Hypercomplex tensor completion via convex optimization, IEEE Trans. Signal Process. 67, 4078–4092.10.1109/TSP.2019.2922156CrossRefGoogle Scholar
Mokhtari, A., Gürbüzbalaban, M. and Ribeiro, A. (2018), Surpassing gradient descent provably: A cyclic incremental method with linear convergence rate, SIAM J. Optim. 28, 1420–1447.10.1137/16M1101702CrossRefGoogle Scholar
Moreau, J. J. (1962), Fonctions convexes duales et points proximaux dans un espace hilbertien, C. R. Acad. Sci. Paris Sér. A Math. 255, 2897–2899.Google Scholar
Moreau, J. J. (1965), Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93, 273–299.10.24033/bsmf.1625CrossRefGoogle Scholar
Moreau, J. J. (1966–1967), Fonctionnelles Convexes, Vol. 2 of Séminaire Jean Leray sur les Équations aux Dérivées Partielles, Collège de France, Paris.Google Scholar
Mosco, U. (1972), Dual variational inequalities, J. Math. Anal. Appl. 40, 202–206.10.1016/0022-247X(72)90043-1CrossRefGoogle Scholar
Motzkin, T. S. and Schoenberg, I. J. (1954), The relaxation method for linear inequalities, Canad. J. Math. 6, 393–404.10.4153/CJM-1954-038-xCrossRefGoogle Scholar
Moudafi, A. (2000), On the regularization of the sum of two maximal monotone operators, Nonlinear Anal. 42, 1203–1208.10.1016/S0362-546X(99)00136-4CrossRefGoogle Scholar
Moudafi, A. and Théra, M. (1999), Proximal and dynamical approaches to equilibrium problems, in Ill-Posed Variational Problems and Regularization Techniques, Vol. 477 of Lecture Notes in Economics and Mathematical Systems, Springer, pp. 187–201.10.1007/978-3-642-45780-7_12CrossRefGoogle Scholar
Nemirovski, A. (2004), Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. Optim. 15, 229–251.10.1137/S1052623403425629CrossRefGoogle Scholar
Neubauer, A. (1988), Tikhonov-regularization of ill-posed linear operator equations on closed convex sets, J. Approx. Theory 53, 304–320.10.1016/0021-9045(88)90025-1CrossRefGoogle Scholar
Nguyen, Q. V. (2017), Forward–backward splitting with Bregman distances, Vietnam J. Math. 45, 519–539.10.1007/s10013-016-0238-3CrossRefGoogle Scholar
O’Connor, D. and Vandenberghe, L. (2014), Primal–dual decomposition by operator splitting and applications to image deblurring, SIAM J. Imaging Sci. 7, 1724–1754.10.1137/13094671XCrossRefGoogle Scholar
Oliveira, D. E., Wolkowicz, H. and Xu, Y. (2018), ADMM for the SDP relaxation of the QAP, Math. Program. Comput. 10, 631–658.10.1007/s12532-018-0148-3CrossRefGoogle Scholar
Papadakis, N., Peyré, G. and Oudet, E. (2014), Optimal transport with proximal splitting, SIAM J. Imaging Sci. 7, 212–238.10.1137/130920058CrossRefGoogle Scholar
Papakonstantinou, J. M. and Tapia, R. A. (2013), Origin and evolution of the secant method in one dimension, Amer. Math. Monthly 120, 500–518.10.4169/amer.math.monthly.120.06.500CrossRefGoogle Scholar
Pascali, D. and Sburlan, S. (1978), Nonlinear Mappings of Monotone Type, Editura Academiei, Bucharest.10.1007/978-94-009-9544-4_3CrossRefGoogle Scholar
Passty, G. B. (1979), Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72, 383–390.10.1016/0022-247X(79)90234-8CrossRefGoogle Scholar
Pathak, R. and Wainwright, M. J. (2020), FedSplit: An algorithmic framework for fast federated optimization, in Advances in Neural Information Processing Systems 33 (Larochelle, H. et al., eds), Curran Associates, pp. 7057–7066.Google Scholar
Peaceman, D. W. and Rachford, H. H. (1955), The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3, 28–41.10.1137/0103003CrossRefGoogle Scholar
Pennanen, T. (2000), Dualization of generalized equations of maximal monotone type, SIAM J. Optim. 10, 809–835.10.1137/S1052623498340448CrossRefGoogle Scholar
Pennanen, T. (2002), A splitting method for composite mappings, Numer. Funct. Anal. Optim. 23, 875–890.10.1081/NFA-120016274CrossRefGoogle Scholar
Pesquet, J.-C. and Repetti, A. (2015), A class of randomized primal–dual algorithms for distributed optimization, J. Nonlinear Convex Anal. 16, 2453–2490.Google Scholar
Pesquet, J.-C., Repetti, A., Terris, M. and Wiaux, Y. (2021), Learning maximally monotone operators for image recovery, SIAM J. Imaging Sci. 14, 1206–1237.10.1137/20M1387961CrossRefGoogle Scholar
Petryshyn, W. V. (1966), On the extension and solution of nonlinear operator equations, Illinois J. Math. 10, 255–274.10.1215/ijm/1256055108CrossRefGoogle Scholar
Phillips, D. L. (1962), A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9, 84–97.10.1145/321105.321114CrossRefGoogle Scholar
Phillips, R. S. (1959), Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90, 193–254.10.1090/S0002-9947-1959-0104919-1CrossRefGoogle Scholar
Pierra, G. (1976), Éclatement de contraintes en parallèle pour la minimisation d’une forme quadratique, in Optimization Techniques: Modeling and Optimization in the Service of Man 2, Vol. 41 of Lecture Notes in Computer Science, Springer, pp. 200–218.10.1007/3-540-07623-9_288CrossRefGoogle Scholar
Pierra, G. (1984), Decomposition through formalization in a product space, Math. Program. 28, 96–115.10.1007/BF02612715CrossRefGoogle Scholar
Polyak, B. T. (1964), Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys. 4, 1–17.10.1016/0041-5553(64)90137-5CrossRefGoogle Scholar
Potter, L. C. and Arun, K. S. (1993), A dual approach to linear inverse problems with convex constraints, SIAM J. Control Optim. 31, 1080–1092.10.1137/0331049CrossRefGoogle Scholar
Powell, M. J. D. (1969), A method for nonlinear constraints in minimization problems, in Optimization (Fletcher, R., ed.), Academic, pp. 283–298.Google Scholar
Qin, X. and An, N. T. (2019), Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets, Comput. Optim. Appl. 74, 821–850.10.1007/s10589-019-00124-7CrossRefGoogle Scholar
Raguet, H. (2019), A note on the forward–Douglas–Rachford splitting for monotone inclusion and convex optimization, Optim. Lett. 13, 717–740.10.1007/s11590-018-1272-8CrossRefGoogle Scholar
Raguet, H. and Landrieu, L. (2015), Preconditioning of a generalized forward–backward splitting and application to optimization on graphs, SIAM J. Imaging Sci. 8, 2706–2739.10.1137/15M1018253CrossRefGoogle Scholar
Raguet, H., Fadili, J. and Peyré, G. (2013), A generalized forward–backward splitting, SIAM J. Imaging Sci. 6, 1199–1226.10.1137/120872802CrossRefGoogle Scholar
Raik, E. (1967), Fejér type methods in Hilbert space, Eesti NSV Tead. Akad. Toimetised Füüs.-Mat. 16, 286–293.Google Scholar
Raik, E. (1969), A class of iterative methods with Fejér-monotone sequences, Eesti NSV Tead. .Akad. Toimetised Füüs.-Mat 18, 22–26.Google Scholar
Reich, H. J. (1961), Functional Circuits and Oscillators, Van Nostrand.Google Scholar
Reich, S., Truong, M. T. and Mai, T. N. H. (2020), The split feasibility problem with multiple output sets in Hilbert spaces, Optim. Lett. 14, 2335–2353.10.1007/s11590-020-01555-6CrossRefGoogle Scholar
Renaud, A. and Cohen, G. (1997), An extension of the auxiliary problem principle to nonsymmetric auxiliary operators, ESAIM Control Optim. Calc. Var. 2, 281–306.10.1051/cocv:1997109CrossRefGoogle Scholar
Robinson, S. M. (1998), A reduction method for variational inequalities, Math. Program. 80, 161–169.10.1007/BF01581724CrossRefGoogle Scholar
Robinson, S. M. (1999), Composition duality and maximal monotonicity, Math. Program. 85, 1–13.10.1007/s101070050043CrossRefGoogle Scholar
Robinson, S. M.(2001), Generalized duality in variational analysis, in Advances in Convex Analysis and Global Optimization (Hadjisavvas, N. and Pardalos, P. M., eds), Kluwer, pp. 205–219.10.1007/978-1-4613-0279-7_10CrossRefGoogle Scholar
Rockafellar, R. T. (1967), Duality and stability in extremum problems involving convex functions, Pacific J. Math. 21, 167–187.10.2140/pjm.1967.21.167CrossRefGoogle Scholar
Rockafellar, R. T. (1969), Convex functions and duality in optimization problems and dynamics, in Mathematical Systems Theory and Economics I (Kuhn, H. W. and Szegö, G. P., eds), Springer, pp. 117–141.Google Scholar
Rockafellar, R. T. (1970a), Convex Analysis, Princeton University Press.10.1515/9781400873173CrossRefGoogle Scholar
Rockafellar, R. T. (1970b), Monotone operators associated with saddle-functions and minimax problems, in Nonlinear Functional Analysis, Part 1 (Browder, F. E., ed.), AMS, pp. 241–250.10.1090/pspum/018.1/0285942CrossRefGoogle Scholar
Rockafellar, R. T. (1971), Saddle-points and convex analysis, in Differential Games and Related Topics (Kuhn, H. W. and Szegö, G. P., eds), North-Holland, pp. 109–127.Google Scholar
Rockafellar, R. T. (1973), The multiplier method of Hestenes and Powell applied to convex programming, J. Optim. Theory Appl. 12, 555–562.10.1007/BF00934777CrossRefGoogle Scholar
Rockafellar, R. T. (1974), Conjugate Duality and Optimization, SIAM.10.1137/1.9781611970524CrossRefGoogle Scholar
Rockafellar, R. T. (1976a), Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1, 97–116.10.1287/moor.1.2.97CrossRefGoogle Scholar
Rockafellar, R. T. (1976b), Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14, 877–898.10.1137/0314056CrossRefGoogle Scholar
Rockafellar, R. T. (1984), Network Flows and Monotropic Optimization, Wiley.Google Scholar
Rockafellar, R. T. (1995), Monotone relations and network equilibrium, in Variational Inequalities and Network Equilibrium Problems (Giannessi, F. and Maugeri, A., eds), Plenum, pp. 271–288.Google Scholar
Rockafellar, R. T. (2024), Generalizations of the proximal method of multipliers in convex optimization, Comput. Optim. Appl. 87, 219–247.10.1007/s10589-023-00519-7CrossRefGoogle Scholar
Rockafellar, R. T. and Sun, J. (2019), Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging, Math. Program. A174, 453–471.10.1007/s10107-018-1251-yCrossRefGoogle Scholar
Rockafellar, R. T. and Wets, R. J. B. (1991), Scenarios and policy aggregation in optimization under uncertainty, Math. Oper. Res. 16, 1–29.10.1287/moor.16.1.119CrossRefGoogle Scholar
Ryu, E. K. (2020), Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting, Math. Program. A182, 233–273.10.1007/s10107-019-01403-1CrossRefGoogle Scholar
Ryu, E. K. and Vũ, B. C. (2020), Finding the forward–Douglas–Rachford-forward method, J. Optim. Theory Appl. 184, 858–876.10.1007/s10957-019-01601-zCrossRefGoogle Scholar
Ryu, E. K., Liu, Y. and Yin, W. (2019), Douglas–Rachford splitting and ADMM for pathological convex optimization, Comput. Optim. Appl. 74, 747–778.10.1007/s10589-019-00130-9CrossRefGoogle Scholar
Ryu, E. K., Taylor, A. B., Bergeling, C. and Giselsson, P. (2020), Operator splitting performance estimation: Tight contraction factors and optimal parameter selection, SIAM J. Optim. 30, 2251–2271.10.1137/19M1304854CrossRefGoogle Scholar
Salzo, S. and Villa, S. (2022), Parallel random block-coordinate forward–backward algorithm: A unified convergence analysis, Math. Program. A193, 225–269.10.1007/s10107-020-01602-1CrossRefGoogle Scholar
Seeger, A. (1998), Alternating projection and decomposition with respect to two convex sets, Math. Japon. 47, 273–280.Google Scholar
Shefi, R. and Teboulle, M. (2014), Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim. 24, 269–297.10.1137/130910774CrossRefGoogle Scholar
Showalter, R. E. (1997), Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society.Google Scholar
Sibony, M. (1970), Méthodes itératives pour les équations et inéquations aux dérivées partielles non linéaires de type monotone, Calcolo 7, 65–183.10.1007/BF02575559CrossRefGoogle Scholar
Sicre, M. R. (2020), On the complexity of a hybrid proximal extragradient projective method for solving monotone inclusion problems, Comput. Optim. Appl. 76, 991–1019.10.1007/s10589-020-00200-3CrossRefGoogle Scholar
Singh, S., Weiss, G. and Tucsnak, M. (2022), A class of incrementally scattering-passive nonlinear systems, Automatica 142, art. 110369.10.1016/j.automatica.2022.110369CrossRefGoogle Scholar
Solodov, M. V. (2004), A class of decomposition methods for convex optimization and monotone variational inclusions via the hybrid inexact proximal point framework, Optim. Methods Softw. 19, 557–575.10.1080/1055678042000218957CrossRefGoogle Scholar
Solodov, M. V. and Svaiter, B. F. (1999a), A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator, Set-Valued Var. .Anal 7, 323–345.10.1023/A:1008777829180CrossRefGoogle Scholar
Solodov, M. V. and Svaiter, B. F. (1999b), A hybrid projection-proximal point algorithm, J. Convex Anal. 6, 59–70.Google Scholar
Solodov, M. V. and Svaiter, B. F. (2000), Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. A87, 189–202.10.1007/s101079900113CrossRefGoogle Scholar
Solodov, M. V. and Svaiter, B. F. (2001), A unified framework for some inexact proximal point algorithms, Numer. Funct. Anal. Optim. 22, 1013–1035.10.1081/NFA-100108320CrossRefGoogle Scholar
Spingarn, J. E. (1983), Partial inverse of a monotone operator, Appl. Math. Optim. 10, 247–265.10.1007/BF01448388CrossRefGoogle Scholar
Spingarn, J. E. (1985), Applications of the method of partial inverses to convex programming: Decomposition, Math. Program. 32, 199–223.10.1007/BF01586091CrossRefGoogle Scholar
Spingarn, J. E. (1987), A projection method for least-squares solutions to overdetermined systems of linear inequalities, Linear Algebra Appl. 86, 211–236.10.1016/0024-3795(87)90296-5CrossRefGoogle Scholar
Steidl, G. and Teuber, T. (2010), Removing multiplicative noise by Douglas–Rachford splitting methods, J. Math. Imaging Vision 36, 168–184.10.1007/s10851-009-0179-5CrossRefGoogle Scholar
Svaiter, B. F. (2011), On weak convergence of the Douglas–Rachford method, SIAM J. Control Optim. 49, 280–287.10.1137/100788100CrossRefGoogle Scholar
Svaiter, B. F. (2014), A class of Fejér convergent algorithms, approximate resolvents and the hybrid proximal-extragradient method, J. Optim. Theory Appl. 162, 133–153.10.1007/s10957-013-0449-7CrossRefGoogle Scholar
Teboulle, M. (1992), Entropic proximal mappings with applications to nonlinear programming, Math. Oper. Res. 17, 670–690.10.1287/moor.17.3.670CrossRefGoogle Scholar
Teboulle, M. (2018), A simplified view of first order methods for optimization, Math. Program. B170, 67–96.10.1007/s10107-018-1284-2CrossRefGoogle Scholar
Tellegen, B. D. H. (1948), The gyrator, a new electric network element, Philips Res. Rept. 3, 81–101.Google Scholar
Thekumparampil, K. K., Jain, P., Netrapalli, P. and Oh, S. (2019), Efficient algorithms for smooth minimax optimization, in Advances in Neural Information Processing Systems 32 (Wallach, H. et al., eds), Curran Associates.Google Scholar
Traoré, C., Salzo, S. and Villa, S. (2023), Convergence of an asynchronous block-coordinate forward–backward algorithm for convex composite optimization, Comput. Optim. Appl. 86, 303–344.10.1007/s10589-023-00489-wCrossRefGoogle Scholar
Tseng, P. (1990), Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Math. Program. 48, 249–263.10.1007/BF01582258CrossRefGoogle Scholar
Tseng, P. (1991), Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim. 29, 119–138.10.1137/0329006CrossRefGoogle Scholar
Tseng, P. (2000), A modified forward–backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38, 431–446.10.1137/S0363012998338806CrossRefGoogle Scholar
Vaĭnberg, M. M. (1956), Variatsionnye Metody Issledovaniya Nelineinykh Operatorov, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow. English translation: Variational Methods for the Study of Non-Linear Operators, Holden-Day (1964).Google Scholar
Vaĭnberg, M. M. (1959), New theorems for non-linear operators and equations, Dokl. Akad. Nauk SSSR 129, 1199–1202.Google Scholar
Vaĭnberg, M. M. (1960), On the convergence of the method of steepest descent for nonlinear equations, Dokl. Akad. Nauk SSSR 130, 9–12.Google Scholar
Vaĭnberg, M. M. (1961), On the convergence of the process of steepest descent for nonlinear equations, Sibirsk. Mat. Zh. 2, 201–220.Google Scholar
Vaĭnberg, M. M. (1972), Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Nauka. English translation: Wiley (1973).Google Scholar
Vaĭnberg, M. M. and Kačurovskiĭ, R. I. (1959), On the variational theory of nonlinear operators and equations, Dokl. Akad. Nauk SSSR 129, 1199–1202.Google Scholar
Vaiter, S., Peyré, G. and Fadili, J. (2018), Model consistency of partly smooth regularizers, IEEE Trans. Inform. Theory 64, 1725–1737.10.1109/TIT.2017.2713822CrossRefGoogle Scholar
Veinott, A. F. (1967), The supporting hyperplane method for unimodal programming, Oper. Res. 15, 147–152.10.1287/opre.15.1.147CrossRefGoogle Scholar
Vese, L. A. and Osher, S. J. (2004), Image denoising and decomposition with total variation minimization and oscillatory functions, J. Math. Imaging Vision 20, 7–18.10.1023/B:JMIV.0000011316.54027.6aCrossRefGoogle Scholar
Villa, S., Rosasco, L., Mosci, S. and Verri, A. (2014), Proximal methods for the latent group lasso penalty, Comput. Optim. Appl. 58, 381–407.10.1007/s10589-013-9628-6CrossRefGoogle Scholar
Vishik, M. I. (1961), Boundary-value problems for quasilinear strongly elliptic systems of equations having divergence form, Soviet Math. Dokl. 2, 643–647.Google Scholar
Vũ, B. C. (2013), A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Comput. Math. 38, 667–681.10.1007/s10444-011-9254-8CrossRefGoogle Scholar
Wang, X., Zhang, J. and Zhang, W. (2020), The distance between convex sets with Minkowski sum structure: Application to collision detection, Comput. Optim. Appl. 77, 465–490.10.1007/s10589-020-00211-0CrossRefGoogle Scholar
Winston, E. and Kolter, J. Z. (2020), Monotone operator equilibrium networks, in Advances in Neural Information Processing Systems 33 (Larochelle, H. et al., eds), Curran Associates, pp. 10718–10728.Google Scholar
Won, J.-H., Xu, J. and Lange, K. (2019), Projection onto Minkowski sums with application to constrained learning, in Proceedings of the 36th International Conference on Machine Learning (Chaudhuri, K. and Salakhutdinov, R., eds), Vol. 97 of Proceedings of Machine Learning Research, PMLR, pp. 3642–3651.Google Scholar
Wright, S. J. and Recht, B. (2022), Optimization for Data Analysis, Cambridge University Press.10.1017/9781009004282CrossRefGoogle Scholar
Xue, F. (2023a), Equivalent resolvents of Douglas–Rachford splitting and other operator splitting algorithms: A unified degenerate proximal point analysis, Optimization. Available at doi:10.1080/02331934.2023.2231005.CrossRefGoogle Scholar
Xue, F. (2023b), A generalized forward–backward splitting operator: Degenerate analysis and applications, Comput. Appl. Math. 42, art. 9.10.1007/s40314-022-02143-3CrossRefGoogle Scholar
Yan, X. and Bien, J. (2021), Rare feature selection in high dimensions, J. Amer. Statist. Assoc. 116, 887–900.10.1080/01621459.2020.1796677CrossRefGoogle Scholar
Yi, P. and Ching, S. (2020), Synthesis of recurrent neural dynamics for monotone inclusion with application to Bayesian inference, Neural Networks 131, 231–241.10.1016/j.neunet.2020.07.037CrossRefGoogle ScholarPubMed
Yoon, T. and Ryu, E. K. (2021), Accelerated algorithms for smooth convex-concave minimax problems with $\mathcal{O}\left(1/{k}^2\right)$ rate on squared gradient norm, in Proceedings of the 38th International Conference on Machine Learning (Meila, M. and Zhang, T., eds), Vol. 139 of Proceedings of Machine Learning Research, PMLR, pp. 12098–12109.Google Scholar
Youla, D. C. (1987), Mathematical theory of image restoration by the method of convex projections, in Image Recovery: Theory and Application (Stark, H., ed.), Academic Press, pp. 29–77.Google Scholar
Yu, Y., Peng, J., Han, X. and Cui, A. (2017), A primal Douglas–Rachford splitting method for the constrained minimization problem in compressive sensing, Circuits Syst. Signal Process. 36, 4022–4049.10.1007/s00034-017-0498-5CrossRefGoogle Scholar
Zames, G. (1966a), On the input–output stability of time-varying nonlinear feedback systems part I: Conditions derived using concepts of loop gain, conicity, and positivity, IEEE Trans. Automat. Control 11, 228–238.10.1109/TAC.1966.1098316CrossRefGoogle Scholar
Zames, G. (1966b), On the input–output stability of time-varying nonlinear feedback systems part II: Conditions involving circles in the frequency plane and sector nonlinearities, IEEE Trans. Automat. Control 11, 465–476.10.1109/TAC.1966.1098356CrossRefGoogle Scholar
Zames, G. and Falb, P. L. (1968), Stability conditions for systems with monotone and slope-restricted nonlinearities, SIAM J. Control 6, 89–108.10.1137/0306007CrossRefGoogle Scholar
Zangwill, W. I. (1969), Nonlinear Programming: A Unified Approach, Prentice Hall.Google Scholar
Zarantonello, E. H. (1960), Solving functional equations by contractive averaging. Mathematical Research Center technical summary report no. 160, University of Wisconsin, Madison.Google Scholar
Zarantonello, E. H. (1964), The closure of the numerical range contains the spectrum, Bull. Amer. Math. Soc. 70, 781–787.10.1090/S0002-9904-1964-11237-4CrossRefGoogle Scholar
Zarantonello, E. H., ed. (1971), Contributions to Nonlinear Functional Analysis, Academic.Google Scholar
Zeidler, E. (1990), Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators, Springer.Google Scholar