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Volumn 5, Issue 1, 2009, Pages 17-37

Augmented Lagrangian and proximal alternating direction methods of multipliers in Hilbert spaces. Applications to games, PDE's and control

Author keywords

Alternating directions method; Augmented Lagrangian; Best response; Convex optimization; Costs to change; Domain decomposition for PDE's; Multipliers; Optimal control; Potential games; Proximal methods; Splitting methods; Variational inequalities; Weak coupling

Indexed keywords


EID: 75249087243     PISSN: 13489151     EISSN: 13498169     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (56)

References (46)
  • 2
    • 0030785144 scopus 로고    scopus 로고
    • On Gabay's algorithms for mixed variational inequalities
    • G. Alduncin, On Gabay's algorithms for mixed variational inequalities, Appl. Math. Optim. 35 (1997) 21-44.
    • (1997) Appl. Math. Optim. , vol.35 , pp. 21-44
    • Alduncin, G.1
  • 4
    • 48549105307 scopus 로고    scopus 로고
    • A new class of alternating proximal minimization algorithms with costs-to-move
    • H. Attouch, P. Redont and A. Soubeyran, A new class of alternating proximal minimization algorithms with costs-to-move, SIAM Journal on Optimization 18 (2007) 1061-1081.
    • (2007) SIAM Journal on Optimization , vol.18 , pp. 1061-1081
    • Attouch, H.1    Redont, P.2    Soubeyran, A.3
  • 5
    • 48549093619 scopus 로고    scopus 로고
    • Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's
    • H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's, J. Convex Anal. 15 (2008) 485-506.
    • (2008) J. Convex Anal. , vol.15 , pp. 485-506
    • Attouch, H.1    Bolte, J.2    Redont, P.3    Soubeyran, A.4
  • 6
    • 33747712643 scopus 로고    scopus 로고
    • Inertia and reactivity in decision making as cognitive variational inequalities
    • H. Attouch and A. Soubeyran, Inertia and reactivity in decision making as cognitive variational inequalities, J. Convex Anal. 13 (2006) 207-224.
    • (2006) J. Convex Anal. , vol.13 , pp. 207-224
    • Attouch, H.1    Soubeyran, A.2
  • 8
    • 41849132222 scopus 로고    scopus 로고
    • Joint minimization with alternating Bregman proximity operators
    • H.H. Bauschke, P.L. Combettes and D. Noll, Joint minimization with alternating Bregman proximity operators, Pacific J. Optim. 2 (2006) 401-424.
    • (2006) Pacific J. Optim. , vol.2 , pp. 401-424
    • Bauschke, H.H.1    Combettes, P.L.2    Noll, D.3
  • 9
    • 10044229235 scopus 로고    scopus 로고
    • The asymptotic behavior of the composition of two resolvents
    • H.H. Bauschke, P.L. Combettes and S. Reich, The asymptotic behavior of the composition of two resolvents, Nonlinear Anal. 60 (2005) 283-301.
    • (2005) Nonlinear Anal. , vol.60 , pp. 283-301
    • Bauschke, H.H.1    Combettes, P.L.2    Reich, S.3
  • 10
    • 0032635368 scopus 로고    scopus 로고
    • Primal-dual strategy for constrained optimal control problems
    • M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim. 37 (1999) 1176-1194.
    • (1999) SIAM J. Control Optim. , vol.37 , pp. 1176-1194
    • Bergounioux, M.1    Ito, K.2    Kunisch, K.3
  • 13
    • 18744361862 scopus 로고    scopus 로고
    • Proximal point algorithm controlled by a slowly vanishing term. Application to hierarchical minimization
    • A. Cabot, Proximal point algorithm controlled by a slowly vanishing term. Application to hierarchical minimization, SIAM J. Optim. 15 (2005) 555-572.
    • (2005) SIAM J. Optim. , vol.15 , pp. 555-572
    • Cabot, A.1
  • 14
    • 0002472542 scopus 로고
    • A proximal-based decomposition method for convex minimization problems
    • G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Math. Program. 64 (1994) 81-101.
    • (1994) Math. Program. , vol.64 , pp. 81-101
    • Chen, G.1    Teboulle, M.2
  • 15
    • 0027541192 scopus 로고
    • The foundations of set theoretic estimation
    • P.L. Combettes, The foundations of set theoretic estimation, Proceedings of the IEEE 81 (1993) 182-208.
    • (1993) Proceedings of the IEEE , vol.81 , pp. 182-208
    • Combettes, P.L.1
  • 17
    • 34249837486 scopus 로고
    • On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators
    • J. Eckstein and D.P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program. 55 (1992) 293-318.
    • (1992) Math. Program. , vol.55 , pp. 293-318
    • Eckstein, J.1    Bertsekas, D.P.2
  • 19
    • 0002788352 scopus 로고
    • Application of the alternating direction method of multipliers to separable convex programming problems
    • M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Comput. Optim. Appl. 1 (1992) 83-111.
    • (1992) Comput. Optim. Appl. , vol.1 , pp. 83-111
    • Fukushima, M.1
  • 20
    • 77957064934 scopus 로고
    • Applications of the method of multipliers to variational inequalities
    • in, M. Fortin, R. Glowinski (eds.), North-Holland, Amsterdam
    • D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrange Methods: Applications to the Solution of Boundary Valued Problems, M. Fortin, R. Glowinski (eds.), North-Holland, Amsterdam, 1983, pp. 299-331.
    • (1983) Augmented Lagrange Methods: Applications to the Solution of Boundary Valued Problems , pp. 299-331
    • Gabay, D.1
  • 21
    • 0002211517 scopus 로고
    • A dual algorithm for the solution of nonlinear variational problems via finite elements approximations
    • D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite elements approximations, Comp. Math. Appl. 2 (1976) 17-40.
    • (1976) Comp. Math. Appl. , vol.2 , pp. 17-40
    • Gabay, D.1    Mercier, B.2
  • 23
    • 0003200255 scopus 로고
    • Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics
    • Philadelphia
    • R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, Philadelphia, 1989.
    • (1989) SIAM Studies in Applied Mathematics
    • Glowinski, R.1    le Tallec, P.2
  • 24
    • 0016536442 scopus 로고
    • Sur l'approximation par éléments finis d'ordre un, et la résolution par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires
    • R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la résolution par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires, Revue Francaise d'Automatique, Informatique et Recherche Opérationnelle 2 (1975) 41- 76.
    • (1975) Revue Francaise d'Automatique, Informatique et Recherche Opérationnelle , vol.2 , pp. 41-76
    • Glowinski, R.1    Marrocco, A.2
  • 25
    • 0032178315 scopus 로고    scopus 로고
    • Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities
    • B.S. He and H. Yang, Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities, Oper. Res. Lett. 23 (1998) 151-161.
    • (1998) Oper. Res. Lett. , vol.23 , pp. 151-161
    • He, B.S.1    Yang, H.2
  • 26
    • 85016616075 scopus 로고    scopus 로고
    • A modified variable-penalty alternating directions method for monotone variational inequalities
    • B.S. He, S.L. Wang and H. Yang, A modified variable-penalty alternating directions method for monotone variational inequalities, J. Comput. Math. 21 (2003) 495-504.
    • (2003) J. Comput. Math. , vol.21 , pp. 495-504
    • He, B.S.1    Wang, S.L.2    Yang, H.3
  • 28
    • 0003053237 scopus 로고
    • An augmented Lagrangian technique for variational inequalities
    • K. Ito and K. Kunisch, An augmented Lagrangian technique for variational inequalities, Appl. Math. Opt. 21 (1990) 223-241.
    • (1990) Appl. Math. Opt. , vol.21 , pp. 223-241
    • Ito, K.1    Kunisch, K.2
  • 29
    • 0000299558 scopus 로고    scopus 로고
    • Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces
    • K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces, Nonlinear Analysis, Theory, Methods and Applications 41 (2000) 573-589.
    • (2000) Nonlinear Analysis, Theory, Methods and Applications , vol.41 , pp. 573-589
    • Ito, K.1    Kunisch, K.2
  • 32
    • 0004232443 scopus 로고
    • Contrôle des systèmes distribués singuliers
    • Gauthiers-Villars, 13, Paris
    • J.L. Lions, Contrôle des systèmes distribués singuliers, Méthodes Mathématiques de l'Informatique, Gauthiers-Villars, 13, Paris, 1983.
    • (1983) Méthodes Mathématiques de l'Informatique
    • Lions, J.L.1
  • 33
    • 28244431909 scopus 로고    scopus 로고
    • Adaptative scaling and convergence rates of a separable augmented Lagrangian algorithm
    • in, V.H. Nguyen, J.J. Strodiot and P. Tossings (eds.), Lectures Notes in Economics and Mathematical Systems, Springer Verlag, Berlin
    • P. Mahey, J.P. Dussault and A. Hamdi, Adaptative scaling and convergence rates of a separable augmented Lagrangian algorithm, in Recent Advances in Optimization, V.H. Nguyen, J.J. Strodiot and P. Tossings (eds.), Lectures Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, 481, 2000, pp. 278-287.
    • (2000) Recent Advances in Optimization , vol.481 , pp. 278-287
    • Mahey, P.1    Dussault, J.P.2    Hamdi, A.3
  • 34
    • 34548678430 scopus 로고    scopus 로고
    • Strong convergence of an iterative method for hierarchical fixed-point problems
    • P.-E. Mainge and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed-point problems, Pacific J. Optim. 3 (2007) 529-538.
    • (2007) Pacific J. Optim. , vol.3 , pp. 529-538
    • Mainge, P.-E.1    Moudafi, A.2
  • 35
    • 0033634301 scopus 로고    scopus 로고
    • Viscosity approximations methods for fixed-points problems
    • A. Moudafi, Viscosity approximations methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46-55.
    • (2000) J. Math. Anal. Appl. , vol.241 , pp. 46-55
    • Moudafi, A.1
  • 36
    • 0041106596 scopus 로고    scopus 로고
    • Finding a zero of the sum of two maximal monotone operators
    • A. Moudafi and M. Théra, Finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl. 94 (1997) 425-448.
    • (1997) J. Optim. Theory Appl. , vol.94 , pp. 425-448
    • Moudafi, A.1    Théra, M.2
  • 38
    • 84968481460 scopus 로고
    • Weak convergence of the sequence of successive approximations for nonexpansive mappings
    • Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc 73 (1967) 591-597.
    • (1967) Bull. Amer. Math. Soc , vol.73 , pp. 591-597
    • Opial, Z.1
  • 39
    • 0003654177 scopus 로고
    • Monotone operators associated with saddle-functions and mini-max problems
    • in, F. Browder (ed.), Proceedings of Symposia in Pure Mathematics, American Mathematical Society
    • R.T. Rockafellar, Monotone operators associated with saddle-functions and mini-max problems, in Nonlinear Functional Analysis, F. Browder (ed.), Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 18, n° 1, 1976, pp. 241-250.
    • (1976) Nonlinear Functional Analysis , vol.18 , Issue.1 , pp. 241-250
    • Rockafellar, R.T.1
  • 40
    • 0016985417 scopus 로고
    • Monotone operators and the proximal point algorithm
    • R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877-898.
    • (1976) SIAM J. Control Optim. , vol.14 , pp. 877-898
    • Rockafellar, R.T.1
  • 41
    • 0016950796 scopus 로고
    • Augmented lagrangians and applications of the proximal point algorithm in convex programming
    • R.T. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research 1 (1976) 97-116.
    • (1976) Mathematics of Operations Research , vol.1 , pp. 97-116
    • Rockafellar, R.T.1
  • 43
    • 0022076946 scopus 로고
    • Applications of the method of partial inverses to convex programming: Decomposition
    • J.E. Spingarn, Applications of the method of partial inverses to convex programming: decomposition, Math. Program. 32 (1985) 199-223.
    • (1985) Math. Program. , vol.32 , pp. 199-223
    • Spingarn, J.E.1
  • 44
    • 0026077016 scopus 로고
    • Applications of splitting algorithm to decomposition in convex programming and variational inequalities
    • P. Tseng, Applications of splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim. 29 (1991) 119-138.
    • (1991) SIAM J. Control Optim. , vol.29 , pp. 119-138
    • Tseng, P.1
  • 45
    • 0031285675 scopus 로고    scopus 로고
    • Alternating projection-proximal methods for convex programming and variational inequalities
    • P. Tseng, Alternating projection-proximal methods for convex programming and variational inequalities, SIAM J. Optim. 7 (1997) 951-965.
    • (1997) SIAM J. Optim. , vol.7 , pp. 951-965
    • Tseng, P.1
  • 46
    • 34248178198 scopus 로고    scopus 로고
    • Proximal alternating directions method for structured variational inequalities
    • M.H. Xu, Proximal alternating directions method for structured variational inequalities, J. Optim. Theory Appl. 134 (2007) 107-117.
    • (2007) J. Optim. Theory Appl. , vol.134 , pp. 107-117
    • Xu, M.H.1


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