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A dual algorithm for the solution of nonlinear variational problems via finite element approximation. (English) Zbl 0352.65034

65K05Mathematical programming (numerical methods)
49K35Minimax problems (optimality conditions)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35A15Variational methods (PDE)
Full Text: DOI
[1] Rockafellar, R. T.: Convex analysis. (1970) · Zbl 0193.18401
[2] Luenberger, D. G.: Introduction to linear and nonlinear programming. (1973) · Zbl 0297.90044
[3] Geoffrion, D.: Duality in nonlinear programming. A simplified applications-oriented development. SIAM rev. 13, 1-37 (1971) · Zbl 0232.90049
[4] Glowinski, R.; Marrocco, A.: Sur l’approximation, par éléments finis d’ordre 1, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non-linéaires. CR hebd. Séanc. acad. Sci., Paris 278, 1649-1652 (1974) · Zbl 0287.65055
[5] Hestenes, M.: Multiplier and gradient methods. Jota 4, No. 5, 303-320 (1969) · Zbl 0174.20705
[6] Powell, M. J. D.: A method for nonlinear optimization in minimization problems. Optimization (1969) · Zbl 0194.47701
[7] Arrow, K. J.; Hurwicz, L.; Uzawa, U.: Studies in non-linear programming. (1968) · Zbl 0091.16002
[8] R. Glowinski and A. Marrocco, Rapport No 74023. Laboratoire d’Analyse Numérique, L.A. 189, Université de Paris VI. Paris France (to appear in R.A.I.R.O.).
[9] Rockafellar, R. T.: The multiplier method of hestenes and powell applied to convex programming. Jota 12, No. 6, 555-562 (1973) · Zbl 0254.90045
[10] D. P. Bertsekas, On the method of multipliers for convex programming. I.E.E.E. Trans. AC-20, 385--388. · Zbl 0301.49023
[11] B. V. Kort and D. Bertsekas, Combined primal-dual and penalty methods for convex programming, SIAM J. (To appear). · Zbl 0332.90035
[12] M. Fortin, Minimization of some non-differentiable functionals by the Augmented Lagrangian method of Hestenes and Powell. To appear.
[13] Moreau, J. J.: Proximité et dualité dans un espace hilbertien. Bull. soc. Math. fr. 93, 273-299 (1965) · Zbl 0136.12101
[14] Luenberger, D. G.: Control problems with kinks. I.E.E.E. trans. 15, 570-575 (1970)
[15] Cea, J.; Glowinski, R.; Nedelec, J. C.: Minimisation de fonctionnelles non différentiables. Conférence on numerical analysis, dundee. Lectures notes in mathematics, 19-38 (1971)
[16] Bertsekas, D. P.: Non-differentiable optimization via approximation. Math. prog. Study 3 (1976)
[17] Lions, J. L.: Quelques méthodes de résolution des problèmes non-linéaires. (1969)
[18] R. Glowinski, J. L. Lions and R. Tremolieres, Analyse numérique des inéquations variationnelles, Dunod, Paris. To appear.
[19] Zienkiewicz: The finite element method in engineering science. (1970)
[20] Ekeland, I.; Temam, R.: Analyse convexe et problèmes variationnels. (1971)
[21] Céa, J.; Glowinski, R.: Sur des méthodes d’optimisation par relaxation. R.a.i.r.o. 3, 5-32 (1973)
[22] Ciarlet, P. G.; Raviart, P. A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element method. The mathematical foundations of the finite element method, 409-474 (1972) · Zbl 0262.65070
[23] Nédelec, J. C.: Approximation par éléments finis des équations de Riccati. Journées eléments finis (1974)
[24] Mosolov, P. P.; Miasnikov, V. P.: Variational methods in the theory of the fluidity of a visco-plastic medium. Pmm 29, 468-492 (1965)
[25] Duvaut, G.; Lions, J. L.: LES inéquations en mécanique et en physique. (1972) · Zbl 0298.73001
[26] Mercier, B.: Approximation par éléments finis et résolution par un algorithme de pénalisation-dualité d’un problème d’élasto-plasticité. CR hebd. Séanc. acad. Sci., Paris 280, 287-290 (1975)
[27] Mercier, B.: A finite element aptroximation of elasto-plastic problem. International computing symposium, 23-31 (1975)
[28] Glowinski, R.; Marrocco, A.: On the solution of a class of non-linear Dirichlet problems by a penalty-duality method and finite element of order one. Proceedings 6th congress IFIP-TC7 (Optimization) (1975) · Zbl 0311.49019
[29] Lemaréchal, C.: An extension of davidon methods to nondifferentiable functions. Math. prog. Study 3 (1976)
[30] C. Lemaréchal, Thèse d’Etat, Paris (To appear).
[31] Bristeau, M. O.: Thèse de 3ème cycle. (1975)
[32] Vanende-Morreeuw, A.: Thèse de 3ème cycle. (1972)
[33] C. Jouron, Resolution numérique du problème des surfaces minima. Archs Ration. Mech. Analysis (To appear).
[34] Rockafellar, R. T.: A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. prog. 5, 354-373 (1973) · Zbl 0279.90035