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A modification of the forward-backward splitting method for maximal monotone mappings
1. | School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing, 210023, China, China |
References:
[1] |
D. P. Bertsekas, "Constrained Optimization and Lagarange Multiplier Method," Academic Press, New York, 1982. |
[2] |
H. G. Chen, "Forward-Backward Splitting Techniques: Theory and Applications," Ph.D. Thesis, Department of Applied Mathematics, University of Washington, Seattle, WA, 1994. |
[3] |
H. G. Chen and R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM Journal on Control and Optimization, 7 (1997), 421-444.
doi: 10.1137/S1052623495290179. |
[4] |
J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms, Mathematical Programming, 83 (1998), 113-123.
doi: 10.1007/BF02680553. |
[5] |
J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[6] |
J. Eckstein and M. C. Ferris, Operator splitting methods for monotone affine variational inequalities, with a parallel application to optimal control, INFORMS Journal on Computing, 10 (1998), 218-235. |
[7] |
J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in "Large Scale Optimization: State of the Art", (eds. W. W. Hager, D. W. Hearn and P. M. Pardalos), Kluwer Academic Publishers, Dordrecht, The Netherlands, (1994), 115-134.
doi: 10.1007/978-1-4613-3632-7_7. |
[8] |
F. Facchinei and J. S. Pang, "Finite-Dimensional Variatiaonal Inequalities and Complementarity Problems," Spring-Verlag, New York, 2003. |
[9] |
D. Gabay, Applications of the method of multipliers to variational inequalities, in "Augemented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems" (eds. M. Fortin and R. Glowinski), North Holland, Amsterdam, (1983), 299-331.
doi: 10.1016/S0168-2024(08)70034-1. |
[10] |
A. A. Goldstein, Convex programming in Hilbert space, Bulletin of the American Mathematical Society, 70 (1964), 709-710.
doi: 10.1090/S0002-9904-1964-11178-2. |
[11] |
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419.
doi: 10.1137/0329022. |
[12] |
B. S. He, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Computational Optimization and Applications, 35 (2006), 19-46.
doi: 10.1007/s10589-006-6442-4. |
[13] |
B. S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 111-128.
doi: 10.1023/A:1013096613105. |
[14] |
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129-143.
doi: 10.1023/A:1013048729944. |
[15] |
K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, Journal on Control and Optimization, 35 (1997), 1142-1168.
doi: 10.1137/S0363012995281742. |
[16] |
B. Lemaire, Coupling optimization methods and variational convergence, in "Trends inMathematical Optimization" (eds. K. H. Hoffman, J. B. Hiriart-Urruty and J. Zowe, C. Lemarechal), Birkhauser-Verlage, Basel, (1988), 163-179.
doi: 10.1007/978-3-0348-9297-1_12. |
[17] |
B. Lemaire, The proximal algorithm, International Series of Numerical Mathematics, 87 (1989), 73-87 |
[18] |
P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Control and Optimization, 16 (1979), 964-979. |
[19] |
Z. Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: a general approach, Annals of Operations Research, 46 (1993), 157-178.
doi: 10.1007/BF02096261. |
[20] |
B. Martinet, Regularisation d'inéquations variationelles par approximations successives, Rev. Francaise Informat Recherche Opérationnelle (French), 4 (1970), 154-159. |
[21] |
G. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal, 29 (1962), 341-346.
doi: 10.1215/S0012-7094-62-02933-2. |
[22] |
B. Martinet, Determination approchée d'un point fixe d'une application pseudocontractante, Comptes rendus de l'Académie des sciences Paris (French), 274 (1972), 163-165. |
[23] |
D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Editura Academiei, Bucharest, 1978. |
[24] |
R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability," Springer-Verlag, New York, 1989.
doi: 10.1007/BFb0089089. |
[25] |
A. Renaud and G. Cohen, Conditioning and regularization of nonsymmetric operators, Journal of Optimization Theory and Applications, 92 (1997), 127-148.
doi: 10.1023/A:1022692114480. |
[26] |
R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970. |
[27] |
R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, New York, 1997. |
[28] |
P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal on Control and Optimization, 29 (1991), 119-138.
doi: 10.1137/0329006. |
[29] |
P. Tseng, On linear convergence of iterative methods for the variational inequality problem, Journal of Computational and Applied Mathematics, 60 (1995), 237-252.
doi: 10.1016/0377-0427(94)00094-H. |
[30] |
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization, 38 (1998), 431-446.
doi: 10.1137/S0363012998338806. |
[31] |
P. Tseng and M. V. Solodov, Modified projection-type methods for monotone variational inequalities, SIAM Journal on Control and Optimization, 34 (1996), 1814-1830.
doi: 10.1137/S0363012994268655. |
[32] |
E. Zeidler, "Nonlinear Monotone Operators, Nonlinear Functional Analysis and its Applications," II/B, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
show all references
References:
[1] |
D. P. Bertsekas, "Constrained Optimization and Lagarange Multiplier Method," Academic Press, New York, 1982. |
[2] |
H. G. Chen, "Forward-Backward Splitting Techniques: Theory and Applications," Ph.D. Thesis, Department of Applied Mathematics, University of Washington, Seattle, WA, 1994. |
[3] |
H. G. Chen and R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM Journal on Control and Optimization, 7 (1997), 421-444.
doi: 10.1137/S1052623495290179. |
[4] |
J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms, Mathematical Programming, 83 (1998), 113-123.
doi: 10.1007/BF02680553. |
[5] |
J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318.
doi: 10.1007/BF01581204. |
[6] |
J. Eckstein and M. C. Ferris, Operator splitting methods for monotone affine variational inequalities, with a parallel application to optimal control, INFORMS Journal on Computing, 10 (1998), 218-235. |
[7] |
J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in "Large Scale Optimization: State of the Art", (eds. W. W. Hager, D. W. Hearn and P. M. Pardalos), Kluwer Academic Publishers, Dordrecht, The Netherlands, (1994), 115-134.
doi: 10.1007/978-1-4613-3632-7_7. |
[8] |
F. Facchinei and J. S. Pang, "Finite-Dimensional Variatiaonal Inequalities and Complementarity Problems," Spring-Verlag, New York, 2003. |
[9] |
D. Gabay, Applications of the method of multipliers to variational inequalities, in "Augemented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems" (eds. M. Fortin and R. Glowinski), North Holland, Amsterdam, (1983), 299-331.
doi: 10.1016/S0168-2024(08)70034-1. |
[10] |
A. A. Goldstein, Convex programming in Hilbert space, Bulletin of the American Mathematical Society, 70 (1964), 709-710.
doi: 10.1090/S0002-9904-1964-11178-2. |
[11] |
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 29 (1991), 403-419.
doi: 10.1137/0329022. |
[12] |
B. S. He, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Computational Optimization and Applications, 35 (2006), 19-46.
doi: 10.1007/s10589-006-6442-4. |
[13] |
B. S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 111-128.
doi: 10.1023/A:1013096613105. |
[14] |
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129-143.
doi: 10.1023/A:1013048729944. |
[15] |
K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, Journal on Control and Optimization, 35 (1997), 1142-1168.
doi: 10.1137/S0363012995281742. |
[16] |
B. Lemaire, Coupling optimization methods and variational convergence, in "Trends inMathematical Optimization" (eds. K. H. Hoffman, J. B. Hiriart-Urruty and J. Zowe, C. Lemarechal), Birkhauser-Verlage, Basel, (1988), 163-179.
doi: 10.1007/978-3-0348-9297-1_12. |
[17] |
B. Lemaire, The proximal algorithm, International Series of Numerical Mathematics, 87 (1989), 73-87 |
[18] |
P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM Journal on Control and Optimization, 16 (1979), 964-979. |
[19] |
Z. Q. Luo and P. Tseng, Error bounds and convergence analysis of feasible descent methods: a general approach, Annals of Operations Research, 46 (1993), 157-178.
doi: 10.1007/BF02096261. |
[20] |
B. Martinet, Regularisation d'inéquations variationelles par approximations successives, Rev. Francaise Informat Recherche Opérationnelle (French), 4 (1970), 154-159. |
[21] |
G. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal, 29 (1962), 341-346.
doi: 10.1215/S0012-7094-62-02933-2. |
[22] |
B. Martinet, Determination approchée d'un point fixe d'une application pseudocontractante, Comptes rendus de l'Académie des sciences Paris (French), 274 (1972), 163-165. |
[23] |
D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Editura Academiei, Bucharest, 1978. |
[24] |
R. R. Phelps, "Convex Functions, Monotone Operators and Differentiability," Springer-Verlag, New York, 1989.
doi: 10.1007/BFb0089089. |
[25] |
A. Renaud and G. Cohen, Conditioning and regularization of nonsymmetric operators, Journal of Optimization Theory and Applications, 92 (1997), 127-148.
doi: 10.1023/A:1022692114480. |
[26] |
R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1970. |
[27] |
R. T. Rockafellar and R. J. B. Wets, "Variational Analysis," Springer-Verlag, New York, 1997. |
[28] |
P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal on Control and Optimization, 29 (1991), 119-138.
doi: 10.1137/0329006. |
[29] |
P. Tseng, On linear convergence of iterative methods for the variational inequality problem, Journal of Computational and Applied Mathematics, 60 (1995), 237-252.
doi: 10.1016/0377-0427(94)00094-H. |
[30] |
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization, 38 (1998), 431-446.
doi: 10.1137/S0363012998338806. |
[31] |
P. Tseng and M. V. Solodov, Modified projection-type methods for monotone variational inequalities, SIAM Journal on Control and Optimization, 34 (1996), 1814-1830.
doi: 10.1137/S0363012994268655. |
[32] |
E. Zeidler, "Nonlinear Monotone Operators, Nonlinear Functional Analysis and its Applications," II/B, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4612-0985-0. |
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