Abstract
In this paper a rather general class of modified Lagrangians is described for which the main results of the duality theory hold. Within this class two families of modified Lagrangians are taken into special consideration. The elements of the first family are characterized by so-called stability of saddle points and the elements of the second family generate smooth dual problems. The computational methods naturally connected with each of these two families are examined. Further a more general scheme is considered which exploits the idea of modification with respect to the problem of finding a root of a monotone operator. This scheme yields a unified approach to convex programming problems and to determination of saddle and equilibrium points as well as expands the class of modified Lagrangians.
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References
E.G. Gol’shtein and N.V. Tret’yakov, “Modified Lagrangian functions”, Economics and Mathematical Methods 10 (3) (1974) 568–591 (in Russian).
E.G. Gol’shtein and N.V. Tret’yakov, “The gradient method of minimization and algorithms of convex programming based on modified Lagrangian functions”, Economics and Mathematical Methods 11 (4) (1975) 730–742 (in Russian).
E.G. Gol’shtein, “The modification method for monotone mappings”, Economics and Mathematical Methods 11 (6) (1975) 1144–1159 (in Russian).
E.G. Gol’shtein, Theory of convex programming, AMS Translation Series (1972), (Translation of a book in Russian edited by “Nauka”, Moscow, 1970).
E.G. Gol’shtein, “The generalized gradient method for determination of saddle points”, Economics and Mathematical Methods 8 (4) (1972) 569–579 (in Russian).
G.D. Maistrovskii, “On the gradient methods for determination of saddle points”, Economics and Mathematical Methods 12 (5) (1976) (in Russian).
B.T. Polyak and N.V. Tret’yakov, “On an iterative method of linear programming and its economic interpretation”, Economics and Mathematical Methods 8 (5) (1972) 740–751 (in Russian).
N.V. Tret’yakov, “The method of penalty prices for convex programming problems”, Economics and Mathematical Methods 9 (3) (1973) 525–540 (in Russian).
R.T. Rockafellar, “A dual approach to solving nonlinear programming problems by unconstrained optimization”, Mathematical Programming 5 (3) (1973) 354–373.
R.T. Rockafellar, “The multiplier method of Hestenes and Powell applied to convex programming”, Journal of Optimization Theory and Applications 12 (6) (1973) 555–562.
B.W. Kort and D.P. Bertsekas, “Multiplier methods for convex programming”, Proceedings 1973 IEEE Conf. on Decision and Control (San Diego, California) 428–432.
B.W. Kort and D.P. Bertsekas, “Combined primal dual and penalty methods for convex programming”, SIAM Journal on Control and Optimization 14 (2) (1976) 268–294
R.T. Rockafellar, “Monotone operators and the proximal point algorithm”, SIAM Journal on Control and Optimization 14 (5) (1976).
R.T. Rockafellar, “Augmented Lagrangians and applications of the proximal point algorithm in convex programming”, Mathematics, of Operations Research 1 (2) (1976) 97–116.
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© 1979 The Mathematical Programming Society
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Gol’shtein, E.G., Tret’yakov, N.V. (1979). Modified Lagrangians in convex programming and their generalizations. In: Huard, P. (eds) Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120845
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DOI: https://doi.org/10.1007/BFb0120845
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-642-00798-9
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